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Question

1+32!+73!+154!+...=?


A

e(e+1)

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B

e(1-e)

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C

e(e-1)

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D

3e

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Solution

The correct option is C

e(e-1)


Explanation for the correct option:

Step 1. Finding Tr term of the given series.

we can write each term of

1+32!+73!+154!+... as

a1=1=21-11!,

a2=32!=22-12!,

a3=73!=23-13! and so on

Tr=2r-1r!

Step 2. Find the sum of Tr,

Trr=1=r=12r-1r! ……..(A)

As we know,

ex=1+x+x22!+x33!+x44!+.....+xrr!+.....

ex=1+r=1xrr!

Step 3. Using the expansion.

Put x=2 and r=1 in exexpansion:

e2=1+r=12rr!

r=12rr!=e2-1........(i)

e1=1+r=11r!

r=11r!=e1-1 ……….(2)

Put the values of equation (1) and (2) in equation (A):

r=1Tr=e2-1-e1-1

=e2-e

=e(e-1)

Hence, Option (C) is Correct.


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