Sequence and Series questions
1035 published questions with worked solutions.
- Let S=2+ 67+ 12 7^2 + 20 7^3 + 30 7^4 + . Then 4S is equal to
- If a_1,a_2,a_3, and b_1,b_2,b_3, are A.P., and a_1=2, a_ 10 =3, a_1b_1=1=a_ 10 b_ 10 , then a_4b_4 is equal to
- Let A_1,A_2,A_3, be an increasing geometric progression of positive real numbers. If A_1A_3A_5A_7= 1 1296 and A_2+A_4= 7 36 ,...
- If n arithmetic means are inserted between a and 100 such that the ratio of the first mean to the last mean is 1:7 and a+n=33,...
- Let a_n _ n=0 ^ be a sequence such that a_0=a_1=0 and a_ n+2 =2a_ n+1 -a_n+1 for all n 0. Then _ n=2 ^ a_n 7^n is equal to
- The sum of the infinite series 1+ 56+ 12 6^2 + 22 6^3 + 35 6^4 + 51 6^5 + 70 6^6 + is equal to
- Consider two G.Ps. 2,2^2,2^3, and 4,4^2,4^3, of 60 and n terms respectively. If the geometric mean of all the 60+n terms is (2)^...
- Suppose a_1,a_2, a_n, be an arithmetic progression of natural numbers. If the ratio of the sum of first five terms to the sum of...
- Let the sum of an infinite G.P., whose first term is a and the common ratio is r, be 5. Let the sum of its first five terms be 98...
- Consider the sequence a_1,a_2,a_3, such that a_1=1, a_2=2 and a_ n+2 = 2 a_ n+1 +a_n for n=1,2,3, If (a_1+ 1 a_2 ) (a_2+ 1 a_3 )...
- Let a_n _ n=0 ^ be a sequence such that a_0=a_1=0 and a_ n+2 =3a_ n+1 -2a_n+1, n 0. Then a_ 25 a_ 23 -2a_ 25 a_ 22 -2a_ 23 a_ 24...
- _ r=1 ^ 20 (r^2+1 )(r!) is equal to
- The sum _ n=1 ^ 21 3 (4n-1)(4n+3) is equal to
- Let 0<z<y<x be three real numbers such that 1x, 1y, 1z are in an arithmetic progression and x, 2y,z are in a geometric...
- The sum of all those terms, of the arithmetic progression 3,8,13, ,373, which are not divisible by 3, is equal to
- Let S=109+ 108 5+ 107 5^2 + + 2 5^ 107 + 1 5^ 108 . Then the value of (16S-(25)^ -54 ) is equal to
- Let a_k and b_k , k N, be two G.P.s with common ratio r_1 and r_2 respectively such that a_1=b_1=4 and r_1<r_2. Let c_k=a_k+b_k,...
- The sum to 20 terms of the series 2 2^2-3^2+2 4^2-5^2+2 6^2- is equal to
- The sum _ k=1 ^ 20 (1+2+3+ +k) is
- The sum, _ n=1 ^7 n(n+1)(2n+1) 4 is
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