Sequences and Series

The ratio of the A.M. and G.M. of two positive numbers a and b is m : n. Show that

a : b = (m + √(m² − n²)) : (m − √(m² − n²))

Let the sum of n, 2n, and 3n terms of an A.P. be S₁, S₂, and S₃ respectively. Show that S₃ = 3(S₂ − S₁).

The sum of three numbers in G.P. is 56. If we subtract 1, 7, 21 from these numbers in that order, we obtain an arithmetic progression. Find the numbers.

Let S be the sum, P the product and R the sum of reciprocals of n terms in a G.P. Prove that P²Rⁿ = Sⁿ.

Given: If a, b, c are in G.P. and a^1/x = b^1/y = c^1/z, prove that x, y, z 

The sum of the first four terms of an A.P. is 56. The sum of the last four terms is 112. If its term is 11, then find the number of terms.

Shamshad Ali buys a scooter for Rs 22000. He pays Rs 4000 cash and agrees to pay the balance in annual installment of Rs 1000 plus 10% interest on the unpaid amount. How much will the scooter cost him?

150 workers were engaged to finish a job in a certain number of days. 4 workers dropped out on second day, 4 more workers dropped out on third day and so on. It took 8 more days to finish the work. Find the number of days in which the work was completed.

Find the sum to n terms: 3×1² + 5×2² + 7×3² + …

Find the sum of the following series up to n terms: 

6 + .66 + .666+….

Find the sum to n terms: 1² + (1² + 2²) + (1² + 2² + 3²) + …

Find the sum of integers from 1 to 100 that are divisible by 2 or 5. 

A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of terms occupying odd places, then find its common ratio. 

A person writes a letter to four of his friends. He asks each one of them to copy the letter and mail to four different persons with instruction that they move the chain similarly. Assuming that the chain is not broken and that it costs 50 paise to mail one letter. Find the amount spent on the postage when 8th set of letter is mailed.

The sum of (m + n)^th and (m − n)^th terms of an A.P. is equal to twice the m^th term.

A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of terms occupying odd places, then find its common ratio. 

The p^th, q^th, and r^th terms of an A.P. are a, b, and c respectively. Show that (q − r)a + (r − p)b + (p − q)c = 0.

Show that

(1² + 2·2² + 3·3² + … + n·(n+1)²) / (1² + 2² + 3² + … + (n+1)²) = (3n + 5) / (3n + 1)

If a, b, c, d are in A.P.; a, b, c are in G.P.; and 1/a, 1/b, 1/c are in A.P., prove that a, b, c are in G.P.

Given: If a, b, c, d are in G.P., prove that (a + b), (b + c), (c + d) are in G.P.