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Question

r=1(2r1)(911)r is equal to

A
45
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B
55
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C
Sum of first nine natural numbers
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D
Sum of first ten natural numbers
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Solution

The correct options are
A 45
C Sum of first nine natural numbers
S=r=1(2r1)(911)r
S=911+3(911)2+5(911)3+.... ...(1)
Multiplying (1) by 911, we get
911S=(911)2+3(911)3+5(911)4+... ...(2)
(1) - (2) gives
(1911)S=911+2(911)2+2(911)3+...
2S11=911+2(911)2⎜ ⎜1(1911)⎟ ⎟
2S11=911+2(911)2(112)
2S11=911+8111=9011
S=902=45
And 1+2+3+..9=9(9+1)2=45

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