Question

If the sum of first 9 terms of an AP is equal to the sum of its first 11 terms, then find the sum of its first 20 terms.

Answer 1

Please refer to the image provided below for your query

Answer 2

Sum of first 20 terms of AP is zero.

Given:

• If the sum of first 9 terms of an AP is equal to the sum of its first 11 terms.

To find :

• Find the sum of its first 20 terms.

Solution:

Formula to be used:

$\boxed{\bf S_n = \frac{n}{2} (2a + (n - 1)d) }$

here,

a: first term of AP

d: Common difference

n: number of terms

Step 1:

Write sum of first 9 terms.

$S_9 = \frac{9}{2} (2a + (9 - 1)d)$

or

$\bf S_9 = 9(a + 4d)...eq1$

Write the sum of first 11 terms

$S_{11}= \frac{11}{2} (2a + (11 - 1)d)$

or

$\bf S_{11}= 11(a + 5d)...eq2$

Step 2:

Equate eq1 and eq2.

$\bf S_{9} = S_{11}$

or

$9(a + 4d) = 11(a + 5d)$

or

$9a + 36d = 11a + 55d$

or

$9a - 11a + 36d - 55d = 0$

or

$- 2a - 19d = 0$

or

$2a + 19d = 0$

or

Multiply both side by 20/2

$\frac{20}{2} (2a + (20 - 1)d) = 0 \times \frac{20}{2}$

or

$\frac{20}{2} (2a + (20 - 1)d) = 0$

or

$S_{20} = \frac{20}{2} (2a + (20 - 1)d)$

or

$\bf \red{S_{20} = 0 }$

Thus,

Sum of first 20 terms of AP is zero.

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