Progression and Series — practice questions

157 curated practice questions from Progression and Series (Mathematics JEE Main, NTA Exams) — 104 easy, 33 medium, 20 hard. Every question includes the final answer free; step-by-step worked solutions with a free account.

• Let f(x) be a polynomial function of second degree. If f(1)=f(-1) and a, b, c are in AP, then f ' (a), f ' (b) and f '…
• Three non-zero real numbers form an AP and the squares of these numbers taken in the same order form a GP. Then, the…
• k and d (d being variable) are the n th term and common difference of an A.P. respectively. If the multiplication of…
• If 19th term of a non-zero A.P. is zero, then its (49th term) : (29th term) is:
• If for an A.P. T 3 = 18 and T 7 = 30 then S 17 is equal to
• If non-zero numbers a, b, c are in HP, and the straight line x a + y b + 1 c =0 always passes through a fixed point…
• The sum of the series 1+3 x+6 x^ 2 +10 x^ 3 + is (where |x| < 1)
• Let a_ 1 b_ 1 c form a GP of common ratio r, where 0<r<1 . If a , 2 ~b , 3 c form an AP, then r equals
• nth term of the sequence a, a + d, a + 2d, ............. is
• If x, 1, z are in AP and x, 2, z are in GP then x, 4, z are in
• The value of n, for which a^ n +b^ n a^ n-1 +b^ n-1 is A.M. between a and b is
• The sum of the infinite series is :
• Sum of the series 4 + 6 + 9 + 13 + 18 + ......... n terms, is
• If a 1 , a 2 , a 3 , ... are in A.P. such that a 1 + a 7 + a 16 = 40, then the sum of the first 15 terms of this A.P…
• The sum of all two digit positive numbers which divided by 7 yield 2 or 5 as remainder is-
• Let a 1 , a 2 , a 3 be any positive real numbers, then which of the following statement is not true?
• The fourth term of the sequence 3, 3 2 , 1, ......... is
• If a_ 1 , a_ 2 , a_ 3 , , a_ n an are in A.P. and a_ 1 +a_ 4 +a_ 7 + +a_ 16 =114 , then a _ 1 + a _ m + a _ 11 + a _ 16…
• In an AP of which a is the first term. If the sum of the first p terms is zero, then the sum of next q terms is
• If x, 2x + 2 and 3x + 3 are first three terms of a G.P., then its 4 th term is
• If 1 a(b+c) , 1 b(c+a) , 1 c(a+b) be in HP, then a, b, c are in
• One side of an equilateral triangle is 24 cm. The mid-points of its sides are joined to form another equilateral…
• In a G.P. of even numbers of terms, the sum of all terms is 5 times the sum of odd terms. The common ratio of the GP is
• Let f (x) be a polynomial of second degree. If f(1)=f(-1) and a, b, c are in AP, then f^ (a) f^ (b) f^ (c) are in
• Between 1 and 31, m arithmetic means are inserted so that the ratio of 7th and (m - 1)th means is 5 : 9 then, the value…
• If S 1 , S 2 and S 3 be respectively the sums of n, 2n and 3n terms of a GP then, the value of 5_ 1 [5_ 3 -5_ 2 ] is
• If sixth term of an H.P. is 1 61 and its tenth term is 1 105 , then first term of that H.P. is :
• Let, f(1)=1 and f(n)=2 _ r=1 ^ n-1 f(r) . Then, _ n=1 ^ m f(n) is equal to
• If a_ k a_ k-1 +a_ k-1 a_ k-2 =2 a_ k a_ k-2 , k 3 and here S_ p = _ k=1 ^ p 1 a_ k and given that S_ 2 p S_ p does not…
• Given below are two statements. STATEMENT - 1: In a GP if the (m + n) th term be p and (m - n) th term be q, then its m…
• Three numbers form an increasing GP. If the middle number is doubled, then the new numbers are in AP. The common ratio…
• If x= _ n=a ^ (-1)^ n ^ 2 n and y= _ n=a ^ ^ 2 n , for 0< < 4 , then :
• If 1 1 3 + 1 2 4 + 1 3 5 + + 1 n(n+2) = n(3 n+p) 2(n+1)(n+2) then is equal to
• The sum of first n terms of the series. 1 – 1 + 1 – 1 + ... is
• If a_ 1 , a_ 2 , a_ 3 , a_ n are in H.P then a_ 1 a_ 2 +a_ 3 + +a_ n a_ 2 a_ 1 +a_ 3 + +a_ n a_ 3 a_ 1 +a_ 2 +a_ 4 +…
• The sum to n terms of the series 1+2 (1+ 1 n )+3 (1+ 1 n )^ 2 + . is given by
• Find the sum of the infinite series 1 2 3 + 1 4 5 + 1 6 7 + :
• In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then, the…
• If S _ n = n P + n ( n -1) 2 Q , where S n denotes the sum of the first n terms of an A.P., then the common difference…
• The sum _ k=1 ^ 20 k 1 2^ k is equal to
• The sum to n terms of the series 1 1.3 + 1 3.5 + 1 5.7 + ..... is
• If 4 GM’s be inserted between 160 and 5, then third GM will be
• Let the positive numbers a,b,c,d be in A.P. Then, abc, abd, acd, bcd are
• If a_ 1 b_ 1 c are three unequal positive quantities in HP, then :
• If a _ 1 , a _ 2 , a _ 3 ( a _ 1 >0 ) are three successive terms of a G.P. with common ratio r, the value of r for…
• If a n be the n th term of an AP and if a 7 = 15, then the value of the common difference that would make a 2 a 7 a 12…
• If the sum of the first n terms of the series 3 + 75 + 243 + 507 + is 435 3 , then n equals:
• 1 + 2.2 + 3.2 2 + 4.2 3 + .... + 100.2 99 equals
• For what value of m, a ^ m+1 + b ^ m+1 a ^ m + b ^ m is the arithmetic mean of 'a' and 'b' ?
• Find the sum to infinity of the series 1^ 2 +2^ 2 x +3^ 2 x ^ 2 +4^ 2 x ^ 3 +
• The sum of an infinite G.P. is 4 and the sum of the cubes of its terms is 192. The common ratio of the original G.P. is
• If a, b, c, d are in G.P. then a n + b n , b n + c n , c n + d n are in
• The product of three numbers in GP is 125 and sum of their product taken in pairs is 87 1 2 . Find the numbers ?
• In the arithmetic progression 2, 5, 8, ... up to 50 terms and 3, 5, 7, 9, ... up to 60 terms, find how many terms are…
• n AM’s are inserted between 2 and 38. If third AM is 14 then n is equal to
• Let the sum of the first n terms of a non-constant A.P., a 1 , a 2 , a 3 , ... be 50 n+ n(n-7) 2 A , where A is a…
• Let H_ 1 , H_ 2 , H_ 3 , , H_ 2 n+1 are in HP then _ i=1 ^ 2 (-1) ( H_ i +H_ i+1 H_ i -H_ i+1 ) is equal to :
• If e^ tin ^ 2 x+ ^ 4 x+ n _ e ^ 2 satisfies the quadratic equation x 2 - 9x + 8 = 0 and 0 < x < / 2 , 0 < sin 2 x < 1…
• The third term of a geometric progression is 4. The product of the first five terms is
• Find the sum of an infinitely decreasing G.P., whose first term is equal to b + 2 and the common ratio is 2/c, where b…
• If A 1 , A 2 be two AM’s and G 1 , G 2 be two GM’s between two numbers a and b, then A_ 1 +A_ 2 G_ 1 G_ 2 is equal to
• Let T_ p be the general term of an A.P whose first term is - 1 2 and common difference is 1, then _ r=1 ^ n 1+T_ r T_…
• If A 1 , A 2 are two AM’s between two numbers a and b, then (2A 1 – A 2 ) (2A 2 – A 1 ) is equal to
• The value of _ n [ 1 1 2 + 1 2 3 + 1 3 4 + + 1 n (n+1) ] is ...
• If a, b, c, d are in GP, then the value of (a-c)^ 2 +(b-c)^ 2 +(b-d)^ 2 -(a-d)^ 2 is
• n arithmetic means are inserted in between x and 2y and then between 2x and y. In case the r th mean in both cases be…
• If a, b, c are in G.P. and a 1/x = b 1/y = c 1/z then x, y, z are in
• A man saves Rs 200 in each of the first three months of his service. In each of the subsequent months his saving…
• If the p th term of an AP is q and the q th term is p, then the r th term is :
• If p th , q th and r th terms of an A.P. are equal to corresponding terms of a G.P. and these terms are respectively x…
• Given below are two statements. One is labelled as Assertion and the other is labelled as Reason. STATEMENT - 1…
• Let a, b and c be the 7 th , 11 th and 13 th terms respectively of a non-constant A.P. If these are also the three…
• _ n=0 ^ ( _ e x )^ n n! is equal to
• If the sides of a triangle are in GP and its largest angle is twice the smallest, then the common ratio 'r' satisfies…
• If 1 + 2 + 3 + ..... + n = 45, then 1 3 + 2 3 + 3 3 +....+ n 3 is
• The first and the n th terms of a GP are a and b respectively and if 'p' is the product of the first n-terms, then p 2…
• Find the sum of the series 5+55+555+ n term 5 .
• If a_ 1 , a_ 2 , a_ 3 , , a_ n+1 are in A.P., then 1 a_ 1 a_ 2 + 1 a_ 2 a_ 3 1 a_ n a_ n+1 , is
• If a_ 1 , a_ 2 , a_ 3 , ......... is an A.P such that a_ 1 +a_ 5 +a_ 10 +a_ 15 +a_ 20 +a_ 24 =225, then a_ 1 +a_ 2 +a_…
• If (x-y) x and (x+y) are in HP, then x 5 e c(y / 2) is equal to
• The product of first n (odd) terms of a G.P. whose middle term is m is
• The harmonic mean of the roots of the equation (5+ 2 ) x^ 2 -(4+ 5 ) x+8+2 5 =0 is
• The first, second and middle term of an AP are a, b, c respectively. Sum of all terms is
• If g 1 , g 2 are two G.M’s between two numbers a and b, then g _ 1 ^ 2 ~g _ 2 + g _ 2 ^ 2 ~g _ 1 is equal to
• If the sum of an infinitely decreasing GP is 3, and the sum of the squares of its terms is 9/2, the sum of the cubes of…
• If sum of first n terms of an A.P. (having positive terms) is given by S_ n = (1+2 T_ n ) (1-T_ n ) where T_ n is the n…
• The sum of first n (odd) terms of an A.P. whose middle term is m is
• If 3+5+7+ + n terms 5+8+11+ .+10 terms =7 then the value of n is
• The sum of integers in between 1 and 100 which are divisible by 2 or 5 is
• Find the sum of following series to infinity 1+ 1+ a 2! + 1+ a + a ^ 2 3! + 1+ a + a ^ 2 + a ^ 3 4! +
• The sum of infinite series ( 1 1.2 + 1 3.4 + 1 5.6 + ) equals
• The sum of all numbers between 100 and 10,000 which are of the form n 3 (n N) is equal to
• Given below are two statements. Let a, r R- 0,1,-1 and n be an even number. STATEMENT - 1: a a r a r^ 2 a r^ n-1 = (a^…
• If first, second and eighth terms of a G.P. are respectively n –4 , n n , n 52 , then the value of n is
• If a 1 , a 2 , a 3 , .......a n be an A.P. of non-zero terms, then the value of 1 a _ 1 a _ 2 + 1 a _ 2 a _ 3 + + 1 a _…
• If x , y , z are in A.P. and tan –1 x , tan –1 y and tan –1 z are also in A.P., then
• Given below are two statements. STATEMENT - 1: x =1111 91 times is composite number. STATEMENT - 2: 91 is composite…
• If r th term of a series is (2r + 1) 2 –r , then sum of its infinite terms is
• Let P r [ array lll x_ r & y_ r & z_ r array ] represents three points for r=1,2,3 where x_ 1 , x_ 2 , x_ 3 ; Y_ 1 , Y_…
• If a_ 1 b_ 1 c are in AP; b, c, d are in GP; c_ 1 d_ 1 e are in HP, then a_ 1 c_ 1 e are in
• If a, b, c are in HP, then the value of b+a b-a + b+c b-c is
• If b+c-a a , c+a-b b , a+b-c c are in A.P. then which of the following is in A.P.
• If a, b, c, d are in G.P., then (a 3 + b 3 ) –1 , (b 3 + c 3 ) –1 , (c 3 + d 3 ) –1 are in
• Let a= (x^ 2 +2 x+3 x R ) and b= _ a 1- ^ 2 then the value of _ r=0 ^ n a^ r b^ n-r
• The n th term of a GP is 128 and the sum of its n terms is 255. If its common ratio is 2 then its first term is
• If l _ n be the n th term of the GP of positive numbers. Let _ n=1 ^ 100 a_ 2 n =x and _ n=1 a _ 2 n-1 = such that …
• Let S n denote the sum of first n terms of an A.P. If S 2n = 3 S n then the ratio S 3n /S n is equal to
• If a 1 , a 2 , a 3 , ................, a n are distinct odd natural numbers not divisible by any prime greater than 5…
• Three numbers are in A.P, such that their sum is 18 and sum of their squares is 158. the greatest among them is
• The sum of the series a – (a + d) + (a + 2d) – (a + 3d) + .... upto (2n + 1) terms is
• 5_ n =1+ 1 2 + 1 z^ 2 + + 1 z^ n-1 ; then the least value of n such that z-5_ n < 1 100 is
• The number of terms of the G.P. 3, 3 2 , 3 4 , needed to obtain a sum of 3069 512 is
• Let S_ k = _ r=1 ^ k ^ -1 ( 6^ r 2^ 2 r+1 +3^ 2 r+1 ) Then _ k S_ k is equal to
• Sum of the series 1 + 4 + 13 + 40 + 121 + ....... 16 terms, is
• Let a_ 1 , a_ 2 , , a_ n be positive real numbers in a G.P. for each n , let A_ n , G_ n , H_ n be respectively, the…
• The sum of numbers lying between 10 and 200 which are divisible by 7 will be:
• If the sum of first two terms of an infinite GP is 1 and every term is twice the sum of all the successive terms, then…
• If α and β are the roots of the equation 375 x 2 - 25x - 2 = 0, then _ n r=1 _ n r=1 ^ n ^ r + _ n _ n ^ n ^ r is equal…
• Let 5= 1 n+1 + 1 2(n+1)^ 2 + 1 3(n+1)^ 3 + and T= 1 n - 1 2 n^ 2 + 1 3 n^ 3 - , then :
• If the roots of the equation a(b-c) x^ 2 +b(c-a) x+c(a-b)=0 are equal, then a_ 1 b_ 1 c are in
• If the roots of the equation x^ 3 -12 x^ 2 +39 x-28=0 are in AP, then the common difference will be
• Let T r be the r th term of an A.P., for r = 1, 2, 3,....... If for some positive integers m, n. We have T_ m = 1 n and…
• The sum of the series 1 1 2 3 4 + 1 2 3 4 5 + n-terms is,
• A number consists of three digits which are in GP, the sum of right hand and left hand digits exceed twice the middle…
• If the sum of an infinite decreasing GP is 3 and the sum of squares of it's terms is 9/2, then the sum of cubes of it's…
• If the first and the nth terms of a G.P. are a and b respectively and P is the product of the first n terms, then P 2 =
• The sum of n terms of the following series 1+(1+ x )+ [1+ x + x ^ 2 )+ . will be
• Sum of infinite terms of series 3+5 1 4 +7 1 4^ 2 + . is
• If S is the sum of the first 10 terms of the series ^ -1 ( 1 3 )+ ^ -1 ( 1 7 )+ ^ -1 ( 1 13 )+ ^ -1 ( 1 21 )+ , then…
• If log 2, log (2 x – 1) and log (2 x + 3) are in AP, then the value of x is given by
• If the arithmetic Mean of two positive numbers a and b (a > b) is twice their geometric Mean, then a : b is
• If a, b, c are in G.P then a^ n , b^ n , c^ n are in
• The sum of first ten terms of an AP is four times the sum of its first five terms, then ratio of first term and common…
• array l If x= (a+ a r + a r^ 2 + ) y= (b- b r + b r^ 2 - ) and z= (c+ c r^ 2 + c r^ 4 + ) then x y z equals array
• The sum of the given infinite series is 1+ 2^ 3 2! + 3^ 3 3! + 4^ 3 4! +
• Let x, y be positive real numbers and m, n positive integers. The maximum value of the expression x^ m y^ (1+x^ 2 m )…
• If 5_ 1 , 5_ 2 , 5_ 3 5_ n are the sum of infinite geometric series whose first terms are 1, 2, 3,......n and common…
• If roots of the equation x 3 – 12 x 2 + 39x – 28 = 0 are in AP, then its common difference is
• If a_ 1 , a_ 2 , a_ 3 , .........., a_ n are in H.P. then a_ 1 a_ 2 +a_ 3 + +a_ n , a_ 2 a_ 1 +a_ 3 + +a_ n , a_ 3 a_ 1…
• The sum to n terms of the series 3 1^ 2 + 5 1^ 2 +2^ 2 + 7 1^ 2 +2^ 2 +3^ 2 + , is
• The product of n positive numbers is unity, then their sum is
• Find the sum of first 24 terms of an AP a 1 , a 2 , a 3 ....., if it is known that a 1 + a 5 + a 10 + a 15 + a 20 + a…
• If S = 1 + a + a 2 +........ to then the value of a is
• If a, b, c are in AP then a ( 1 b + 1 c ), b ( 1 c + 1 a ), c ( 1 a + 1 b ) are in
• If the sum of first p terms, first q terms and first r terms of an A.P. be x , y and z respectively, then x p (q-r)+ y…
• The first two terms of a geometric progression add up to 12. The sum of the third and the fourth term is 48. If terms…
• If t _ n = 1 4 ( n +2)( n +3) for n = 1, 2, 3, .........., then 1 t_ 1 + 1 t_ 2 + 1 t_ 3 + + 1 t_ 2003 =
• If 0 a , b –1 a + tan –1 b = 4 , then the value of (a+b)- ( a^ 2 +b^ 2 2 )+ ( a^ 3 +b^ 3 3 )- ( a^ 4 +b^ 4 4 )+ is
• STATEMENT - 1 (Assertion): If x^ 2 +9 y^ 2 +25 z^ 2 =x y z ( 15 x + 5 y + 3 z ) . then x, y, z are in HP. STATEMENT - 2…
• If a_ 1 , a_ 2 , a_ 3 a_ n be an AP of non-zero terms then 1 a_ 1 a_ 2 + 1 a_ 2 a_ 3 + + 1 a_ n-1 a_ n is
• A stick of length 20 units is to be divided into n parts so that the product of the lengths of the part is greater than…
• The fourth, seventh and tenth terms of a G.P. are p, q, r respectively, then
• Let a_ 1 , a_ 2 , a_ 3 , ., a_ n be a GP such that a _ 4 a _ 6 = 1 4 and a_ 2 +a_ 5 =216 . Then integral value of a 1 is
• The largest term common to the sequences 1, 11, 21, 31, .... to 100 terms and 31, 36, 41, 46 ....to 100 terms is
• Let a_ 1 , a_ 2 , , a_ 1 be in AP and h_ 1 , h_ 2 , ., h_ 1 , be in HP. If a _ 1 = h _ 1 = Z and a_ 1! =h_ 1! =3 , then…
• Which of the following sequences is an A.P. with common difference 3 ?
• The sum of n terms of the series 4 3 + 10 9 + 28 27 + is :

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