Every question,
solved step by step.

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?

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?

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.

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 GP is \(\frac{13}{12}\) and their product is −1. Find the common ratio and the terms.

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.

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

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.

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?

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