End Of Chapter Exercises

Find the \(31^{st}\) term of the AP whose \(11^{th}\) term is 38 and \(16^{th}\) term is 73.

Determine the AP whose third term is 16 and whose \(7^{th}\) term exceeds the \(5^{th}\) term by 12.

How many three-digit numbers are divisible by 7? (Hint: All three-digit numbers divisible by 7 form an AP. Find the smallest and largest such three-digit numbers.)

How many multiples of 4 lie between 10 and 250? (Hint: All multiples of 4 form an AP. Find the smallest and largest multiples of 4 between 10 and 250.)

Find a GP for which the sum of the first two terms is −4 and the fifth term is 4 times the third term.

Find all possible ways of expressing 100 as the sum of consecutive natural numbers.

The number of bacteria in a certain culture doubles every hour. If there were 30 bacteria present in the culture originally, how many bacteria will be present at the end of the \(2^{nd}\) hour, \(4^{th}\) hour and \(n^{th}\) hour?

The sum of the \(4^{th}\) and \(8^{th}\) term of an AP is 24 and the sum of the \(6^{th}\) and \(10^{th}\) terms is 44. Find the first three terms of the AP.

Find the smallest value of \(n\) such that the sum of the first \(n\) natural numbers is greater than 1000.

Which term of the GP: 2, 8, 32, ... is 131072? Write the explicit formula as well as the recursive formula for the \(n^{th}\) term.

The sum of the first three terms of a GP is \(\frac{13}{12}\) and their product is −1. Find the common ratio and the terms.

If the \(4^{th}\), \(10^{th}\) and \(16^{th}\) terms of a GP are \(x,\,y\) and \(z\) respectively, prove that \(x,\,y,\,z\) are in GP.

The sum of the first three terms of a geometric progression is 26, and the sum of their squares is 364. Find the terms of the GP.

Suppose \(P_1 = 1,\, P_2 = 2\) and for \(n>2,\,P_n = P_1 + P_2 + ... + P_{n−1} + 1\). Find the values of \(P_1, P_2, ..., P_8\). Can you find a simpler recursive formula for \(P_n\)? Can you give an explicit formula?

Suppose \(W_1 = 1, W_2 = 2\) and for \(n>2, \,W_n = W_1 + W_2 + ... + W_{n−2} + 2\). Find the values of \(W_1, W_2, ..., W_8\). Do you recognise this sequence?