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.
The answer is in the photo, please tell me the steps, thank you​

Answer 1

• See below

Explanation:

Given sequence:

• log x + log√2, 2logx + log 2, 3logx + log2√2, ...

a)

The terms can be rewritten as:

• T(1) = log x + log√2 = log x√2
• T(2) = 2logx + log 2 = 2logx + 2 log√2 = 2log x√2
• T(3) = 3logx + log2√2 = 3logx + 3log√2 = 3log x√2
• ...
• T(n) = n log x√2 (option A)

b)

Sum of the first n terms:

• log x√2 + 2log x√2 + ... + nlog x√2 =
• log x√2(1 + 2 + ... + n) =
• log x√2 (1 + n)*n/2 =
• 1/2n(n + 1)log x√2

Sum of the first 40 terms:

• 1/2*40(40 + 1)log x√2 =
• 820log x√2