In an arithmetic progression, the sum of the first half of the terms is 54,

Question

the sum of the second half is 108 and the sum of the first?

 

1.) in an arithmetic progression, the sum of the first half of the terms is 54, the sum of the second half is 108 and the sum of the first and last term is 54. Find the number of terms.
2.) the first and the last term of a geometric progression is equal to -8 and 1 respectively. If the sum of all the terms is -5, find the number of terms.

Answer 1

Note :

The sum S_n of the first n terms of an arithmetic series is given by , where a₁ = first term, a_n = last term.

The sum S_n of the first n terms of geometric series is given by  , where a = first term, r = common ratio.

1).

Sum of the first half of the terms .

Sum of the second half of the terms .

Sum of the first term and last term .

Total sum = Sum of the first half of the terms + Sum of the first half of the terms.

.

The number of terms in the given arithmetic progression is 6.

Answer 2

2).

First term of the geometric progression (a ) = - 8.

Last term of the geometric progressin  .

Substitute a = - 8 in .

.

Sum of geometric progression .

Substitute a = - 8 and in .

.

To find n, substitute the value r = - 1 / 2 in .

 

⇒ n = 4.

The number of terms of the geometric progression is 4.