Question

Consider the arithmetic sequences 3, 5, 7, ….. and 4, 6, 8, ….. How much more is the sum of the first 25 terms of the second than the sum of the first 25 terms of the first?

Answer 1

Consider the first arithmetic sequence 3, 5, 7, …

First term = a + b = 3

Common difference = a = 5 − 3 = 2

⇒ 2 + b = 3

⇒ b = 3 − 2 = 1

Thus, the n^th term of the first arithmetic sequence is:

We know that the sum of a specified number of the consecutive terms of an arithmetic sequence is half the product of the number of the terms with the sum of the first and the last term.

∴ Sum of its first 25 terms = … (1)

Here,

Putting the values in equation (1):

Sum of first 25 terms of the first arithmetic sequence

Now, consider the second arithmetic sequence 4, 6, 8, …

First term = a + b = 4

Common difference = a = 6 − 4 = 2

⇒ 2 + b = 4

⇒ b = 4 − 2 = 2

Thus, the n^th term of the first arithmetic sequence is:

Sum of its first 25 terms = … (1)

Here,

Putting the values in equation (1):

Sum of first 25 terms of the second arithmetic sequence =

∴ Required difference = 700 − 675 = 25

Thus, the sum of first 25 terms of the second arithmetic sequence is 25 more than the sum of first 25 terms of the first arithmetic sequence.