Question

How many terms of the A.P. -6, -11/2, -5, …. are needed to give the sum 37.5?​

Answer 1

Given,

AP = -6, -11/2, -5

Sum = 37.5

To find,

The number of terms of the given A.P, that is necessary to obtain the given value of sum.

Solution,

We can simply solve this mathematical problem by using the following mathematical process.

First term (a) = -6

Second term = -11/2 = -5.5

Common difference (d) = Second term - First term = -5.5 - (-6) = -5.5+6 = 0.5

Let, the number of terms we need = n

So, according to the formula.

Sum of the n terms = n/2 [2a+(n-1)×d] = n/2 [2×(-6)+(n-1)×0.5]

According to the data mentioned in the question,

n/2 [2×(-6)+(n-1)×0.5] = 37.5

n/2 (-12+n/2-0.5) = 37.5

n/2 (-12.5+n/2) = 37.5

-12.5n/2 + n²/4 = 37.5

-25n+n²/4 = 37.5

-25n+n² = 150

n²-25n-150 = 0

(n+5) (n-30) = 0

So,

Either, n+5 = 0

n = -5

Or, n-30 = 0

n = 30

Now, number of terms will be a positive value, so we will consider n = 30 as the correct answer.

Hence, we need 30 terms.