Question

27. योगफल ज्ञात कीजिए :
(1/2) + (1/6) + (1/12) + ...... + 1/156

Answer 1

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$\frac{1}{2} + \frac{1}{6} + \frac{1}{12} + ............ + \frac{1}{156}$

$= \frac{1}{1 \times 2} + \frac{1}{2 \times 3} + \frac{1}{3 \times 4} + ......... + \frac{1}{12 \times 13}$

$= \frac{(2 - 1)}{1 \times 2} + \frac{(3 - 2)}{2 \times 3} + \frac{(4 - 3)}{3 \times 4} + ......... + \frac{(13 - 12)}{12 \times 13}$

$= \frac{2}{ 1\times 2} - \frac{1}{1 \times 2} + \frac{3}{2 \times 3} - \frac{2}{2 \times 3} + \frac{4}{3 \times 4} - \frac{3}{3 \times 4} + ......... + \frac{13}{12 \times 13} - \frac{12}{12 \times 13}$

$= 1 - \frac{1}{ 2} + \frac{1}{2 } - \frac{1}{ 3} + \frac{1}{3 } - \frac{1}{4} + ......... + \frac{1}{12} - \frac{1}{13}$

$= 1 - \frac{1}{13}$

$= \frac{(13 - 1)}{13}$

$= \frac{12}{13}$

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Answer 2

Step-by-step explanation:

Look for the pattern in the given series. We see it is not a Geometric Progression. Let’s see what else we can find.

N = 1/2 + 1/6 + 1/12 + 1/20 + … + 1/156

N = 1/1*2 + 1/2*3 + 1/3*4 + 1/4*5 + … + 1/12*13

This tells us that we can split each term into two with different denominators. We should be able to cancel the terms to quickly arrive at the answer.

1/2 = 1 - 1/2

1/6 = 1/2 - 1/3

1/12 = 1/3 - 1/4

…

1/156 = 1/12 - 1/13

Therefore,

N = (1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4) + … (1/11 - 1/12) + (1/12 - 1/13)

Notice the consecutive positive and negative terms. They will cancel each other off (-1/2) will cancel off (1/2), (-1/3) will cancel off (1/3) and so on till we are left with the first term 1 and the last term -1/13.

N = 1 - 1/13 = 12/13

Answer = 12/13 (assuming you were looking for the sum of the terms)