Question

In an AP with first term 2. the sum of first 10 terms is equal to the sum of next 4 terms. Find the common difference of the AP​

Answer 1

SOLUTION

GIVEN

In an AP with first term 2. The sum of first 10 terms is equal to the sum of next 4 terms.

TO DETERMINE

The common difference of the AP

CONCEPT TO BE IMPLEMENTED

If in an arithmetic progression

First term = a and common difference = d

Then the sum of first n terms

$= \displaystyle \sf{ \frac{n}{2} \bigg[2a + (n - 1)d\bigg]}$

EVALUATION

Here it is given that

First term = a = 2

Let Common Difference = d

Again it is given that

The sum of first 10 terms is equal to the sum of next 4 terms.

So

Sum of first 10 terms = Sum of first 14 terms - Sum of first 10 terms

Which gives

2 × Sum of first 10 terms = Sum of first 14 terms

$\implies \displaystyle \sf{2 \times \frac{10}{2} \bigg[4 + (10 - 1)d\bigg]} = \frac{14}{2} \bigg[4 + (14 - 1)d\bigg]$

$\implies \displaystyle \sf{10\bigg[4 + 9d\bigg]} = 7 \bigg[4 + 13d\bigg]$

$\implies \displaystyle \sf{40 + 90d} = 28 + 91d$

$\implies \sf{91d - 90d = 40 - 28}$

$\implies \sf{d = 12}$

Hence the Common Difference of the Arithmetic progression = 12

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