Question

How many terms of AP - 6. - 11 by 2 .minus 5 are needed to be the sum -25​

Answer 1

Given:

The terms of a sequence are:

`-6,-(11)/(2),-5`

To find:

The number of terms whose sum is -25.

Solution:

We have, the given sequence:

`-6,-(11)/(2),-5`

Here, the first term is -6.

`-(11)/(2)-(-6)=-5.5+6`

`-(11)/(2)-(-6)=0.5`

Similarly,

`-5-(-(11)/(2))=-5+5.5`

`-5-(-(11)/(2))=0.5`

The difference between consecutive terms are same. So, the given sequence is an arithmetic sequence with common difference 0.5.

The sum of n terms of an arithmetic sequence is:

`S_n=(n)/(2)[2a+(n-1)d]`

Where, a is the first term, d is the common difference.

Putting
`S_n=-25,a=-6,d=0.5`, we get

`-25=(n)/(2)[2(-6)+(n-1)0.5]`

`-50=n[-12+0.5n-0.5]`

`-50=-12.5n+0.5n^2`

`0=0.5n^2-12.5n+50`

Splitting the middle term, we get

`0.5n^2-2.5n-10n+50=0`

`0.5n(n-5)-10(n-5)=0`

`(0.5n-10)(n-5)=0`

Using zero product property, we get

`(0.5n-10)=0` and
`(n-5)=0`

`n=(10)/(0.5)` and
`n=5`

`n=20` and
`n=5`

Therefore, the sum of either 5 or 20 terms is -25.