Question

Consider the arithmetic sequence log x + log √2, 2 log x + log 2, 3 log x + log 2√2......

(a) Find the general term of the sequence.
(b) Find the sum of the first 40 terms of the sequence.​

Answer 1

a)
`t_n = (log x + log √2)n`

b)
`S_(40) =20\bigg(log x + log \sqrt 2\bigg)[ 1+n]`

Explanation:

Given arithmetic sequence is:

log x + log √2, 2 log x + log 2, 3 log x + log 2√2......

First term a = log x + log √2

common difference d = 2 log x + log 2 - (log x + log √2)

= 2 log x + log 2 - log x - log √2

= log x + log 2 - log √2

= log x + log (√2*√2)- log √2

= log x + log √2 + log √2- log √2

= log x + log √2

(a)

General term of the arithmetic sequence is given as:

`t_n = a + (n - 1)d`

`\implies t_n = (log x + log \sqrt 2) + (n - 1)( log x + log √2 )`

`\implies t_n = (log x + log \sqrt 2)[1+ + (n - 1)]`

`\implies t_n = (log x + log \sqrt 2)[1+ n - 1]`

`\implies t_n = (log x + log \sqrt 2)[n]`

`\implies \red{\bold{t_n = (log x + log \sqrt 2)n}}`

(b)

Sum of first n terms of an arithmetic sequence is given as:

`S_n =(n)/(2)\bigg(a+t_n\bigg)`

`\implies S_(40) =(40)/(2)\bigg[ (log x + log \sqrt 2)+(log x + log \sqrt 2)n\bigg]`

`\implies S_(40) =20\bigg(log x + log \sqrt 2\bigg)(1+n)`