Find the value of 12 - 22 + 32 - 42 + 52……… - 20102 + 20112
The required solution can be obtained using the following approaches
Application of the identity a2 - b2 = (a + b)(a - b)
12 - 22 + 32 - 42 + 52……… - 20102 + 20112
= 20112 - 20102…. +52 - 42 + 32 - 22 + 12
= (2011 - 2010)(2011 + 2010) +……..+ (3 - 2)(3 + 2) + 1
= (1).(2011 + 2010)+…….+ (1).(3 + 2)+1
= 2011 + 2010 + …….. +3 +2 +1
Method of generalisation
12 - 22 + 32 = 1 + 2 + 3
12 - 22 + 32 - 42 + 52 = 1 + 2 + 3 + 4 + 5
Hence it follows that 12 - 22 + 32 - 42 + 52.... - 20102 + 20112
is equal to 1 + 2 + 3 + 4 + 5 +…..+ 2010 + 2011
motivation to think of simplifications in the deeper sense of the word came from Prof.N Ch Pattabhiramacharyulu my teacher at NIT-warangal
ReplyDelete