Sequence/series

If the square of the 7th term of an arithmetic progression with positive common difference equals the product of the 3rd and 17th terms, then the ratio of the first term to the common difference is

Let a₁, a₂,...., a_3n be an arithmetic progression with a₁ = 3 and a₂ = 7. If a₁ + a₂...a_3n = 1830, then what is the smallest positive integer m such that m(a1 + a2...an) > 1830?

Let x, y, z be three positive real numbers in a geometric progression such that x < y < z. If 5x, 16y, and 12z are in an arithmetic progression then the common ratio of the geometric progression is

Let a₁, a₂, ... , a₅₂ be positive integers such that a₁ < a₂ < ... < a₅₂. Suppose, their arithmetic mean is one less than the arithmetic mean of a₂, a₃, ..., a₅₂. If a₅₂ = 100, then the largest possible value of a₁ is

Let t₁, t₂,… be real numbers such that t₁+t₂+…+t_n = 2n²+9n+13, for every positive integer n ≥ 2. If t_k=103, then k equals

The value of the sum 7 x 11 + 11 x 15 + 15 x 19 + ...+ 95 x 99 is

If a₁ + a₂ + a₃ + ... + a_n = 3(2_n+1 – 2), for every n ≥ 1, then a₁₁ equals

If (2n + 1) + (2n + 3) + (2n + 5) ... + (2n + 47)
= 5280, then what is the value of 1 + 2 + 3 + ... + n?

The number of common terms in the two sequences: 15, 19, 23, 27, . . . . , 415 and 14, 19, 24, 29, . . . , 464 is

If x₁ = –1 and x_m = x_m+1 + (m + 1) for every positive integer m, then x₁₀₀ equals

The natural numbers are divided into groups as (1), (2, 3, 4), (5, 6, 7, 8, 9), ….. and so on. Then, the sum of the numbers in the 15th group is equal to

Three positive integers x, y and z are in arithmetic progression. If y − x > 2 and xyz =5(x + y + z), then z − x equals

On day one, there are 100 particles in a laboratory experiment. On day n, where n ≥ 2, one out of every n particles produces another particle. If the total number of particles in the laboratory experiment increases to 1000 on day m, then m equals.

A lab experiment measures the number of organisms at 8 am every day. Starting with 2 organisms on the first day, the number of organisms on any day is equal to 3 more than twice the number on the previous day. If the number of organisms on the nth day exceeds one million, then the lowest possible value of n is

Let a_n and b_n be two sequences such that a_n = 13 + 6(n – 1) and b_n = 15 + 7(n – 1) for all natural numbers n. Then, the largest three digit integer that is common to both these sequences, is

Let a_n = 46 + 8_n and bn = 98 + 4n be two sequences for natural numbers n ≤ 100. Then, the sum of all terms common to both the sequences is