Question

Prove that the sum of the first five terms of any arithmetic sequence is five times the middle number. What about the sum of the first seven terms? Can you formulate a general principle from these examples?

Answer 1

(1)

Let the first five terms of an arithmetic sequence be a + b, 2a + b, 3a + b, 4a + b and 5a + b.

Here, the first term = a + b

Common difference = a

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 5 terms = … (1)

Here,

Putting the values in equation (1):

Sum of first 5 terms =

= Five times the middle number, i.e., the third number, 3a + b

Therefore, the sum of the first five terms is five times the middle term.

(2)

Let the first seven terms of an arithmetic sequence be a + b, 2a + b, 3a + b, 4a + b, 5a + b, 6a + b and 7a + b.

Here, first term = a + b

Common difference = a

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 7 terms = … (1)

Here,

Putting the values in equation (1):

Sum of first 7 terms

= Seven times the middle number, i.e., the fourth number, 4a + b

Therefore, the sum of the first seven terms is seven times the middle term.

General principle says that the sum of first n terms of an arithmetic sequence, where n is an odd number is n times the middle number.