Q. 18

# 1 + 2+ 3 +…. + n < ( 2n + 1)2

Let P(n) = 1 + 2+ 3 +…. + n < ( 2n + 1)2
P(1) is true
Let us assume p(k) is true. 1 + 2+ 3 +…. + k < ( 2k + 1)2
To prove P(k+1) is true using P(k)
P(k+1) = 1 + 2+ 3 +…. + k +k+1< ( 2(k + 1)+1)2

= 1+ (1 + 2+ 3 +…. + k) +k < ( 2k + 3)2…….. 1
L .H.S 1+ ( 2k + 1)2 + k
= = ( 4k2 + 1 +4k + 8k + 8)
= (4k2 + 12k +9)
= (2k + 3)2
Which is the R.H.S of……. 1 P(K+1) is true. By the Principle of mathematical induction, P(n) is true for all values of n where n N
Hence proved.

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