Second term of a GP is 1000 and the common ratio is r = 1/n where n is a natural number. P_n is the product of n terms of this GP. P₆ > P₅ and P₆ > P₇, what is the sum of all possible values of n?

Sum of first 12 terms of a GP is equal to the sum of the first 14 terms in the same GP. Sum of the first 17 terms is 92, what is the third term in the GP?

Sum of first 25 terms in AP is 525, sum of the next 25 terms is 725, what is the common difference?

Let the n^th term of AP be defined as t_n, and sum up to 'n' terms be defined as S_n. If |t₈| = |t₁₆| and t₃ is not equal to t₇, what is S₂₃?

a, b, c, d and e are 5 distinct numbers that from an arithmetic progression. They are not necessarily consecutive terms but form the first 5 terms of the AP. It is known that c is the arithmetic mean of a and b, and d is the arithmetic mean of b anc c. Which of the following statements are true?
i. Average of all 5 terms put together is c.
ii. Average of d and e is not greater than average of a and b.
iii. Average of b and c is greater than average of a and d.