Question

Given that for an arithmetic sequence, the first term is q and the second term is qz.

(a) Find the common difference d the general term Tn and the sum of the first n terms. Answer must

be in terms of q and z.
(b) Hence, find the sum of the first 10 terms in terms of q and z. ​

Answer 1

`d = q(z-1)`

`S_n = (nq)/(2)(2 + (n-1)(z-1))`

`S_(10) = 5q(9z -7)`

Explanation:

Given

`T_1 = q`

`T_2 = qz`

Solving (a1): The common difference (d)

d is calculated as

`d = T_2 - T_1`

This gives:

`d = qz - q`

Factorize:

`d = q(z-1)`

Solving (a2): Sum of n terms

This is calculated using:

`S_n = (n)/(2)(2*T_1 + (n-1)d)`

Substitute values for T1 and d

`S_n = (n)/(2)(2*q + (n-1)q(z-1))`

`S_n = (n)/(2)(2q + q(n-1)(z-1))`

Factorize:

`S_n = (n)/(2)(q(2 + (n-1)(z-1)))`

`S_n = (nq)/(2)(2 + (n-1)(z-1))`

Solving (b): Sum of first 10.

In this case, n = 10

So:

`S_n = (nq)/(2)(2 + (n-1)(z-1))`

becomes

`S_(10) = (10 * q)/(2)(2 + (10-1)(z-1))`

`S_(10) = 5 * q(2 + 9(z-1))`

`S_(10) = 5q(2 + 9(z-1))`

`S_(10) = 5q(2 + 9z-9)`

`S_(10) = 5q(2 -9+ 9z)`

`S_(10) = 5q(-7+ 9z)`

`S_(10) = 5q(9z -7)`