Question

if first term of an AP be 100 and the sum of first six terms is five times the sum of next six terms .find the sum of first eleven terms

Answer 1

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Answer 2

Sum of first eleven terms is 550.

Step-by-step explanation:

Given:

First term of an AP = a = 100

Sum of first six terms = 100+(100+d)+(100+2d)+(100+3d)+(100+4d)+(100+5d)

And the sum of next six terms = (100+6d)+(100+7d)+(100+8d)+(100+9d)+(100+10d)+(100+11d)

Here we have,

the sum of first six terms = five times the sum of next six terms

⇒ 100+(100+d)+(100+2d)+(100+3d)+(100+4d)+(100+5d) = 5[(100+6d)+(100+7d)+(100+8d)+(100+9d)+(100+10d)+(100+11d)]

⇒600+15d=5(600+51d)

⇒-240d=2400

⇒$d=-\frac{2400}{240}$

⇒ d=-10

Now,

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

$S_{11} =\frac{11}{2} [2a+(11-1)d]$                      [eleven term, n=11]

$S_{11}=\frac{11}{2} [2(100)+(11-1)d]$                 [ a=100]

$S_{11}=\frac{11}{2} [200+10d]$  

$S_{11}=\frac{11\times 200}{2} +\frac{10\times 11\times d}{2}$

$S_{11}=11\times100+5\times11\times d$

$S_{11}=1100+55(-10)$                               [d=-10]

$S_{11}=1100-550$

$S_{11}=550$