98 Powers of Two

Find the value of the following expression:
\left (98^2-97^2+96^2-95^2+\cdots+2^2-1^2\right )^2
Source: mathcontest.olemiss.edu 4/9/2012

SOLUTION
We simplify the expression by using the identity a^2-b^2=\left (a+b\right )\left (a-b\right )
\left [\left (98+97\right )\left (98-97\right )+\left (96+95\right )\left (96-95\right )+\left (94+93\right )\left (94-93\right )+\cdots+\left (4+3\right )\left (4-3\right )+\left (2+1\right )\left (2-1\right )\right ]^2
\left [\left (195\right )\left (1\right )+\left (191\right )\left (1\right )+\left (187\right )\left (1\right )+\cdots+\left (7\right )\left (1\right )+\left (3\right )\left (1\right )\right ]^2
\left (195+191+187+\cdots+7+3\right )^2
We calculate S=195+191+187+\cdots+7+3 as follows
3=3
7=3+1\left (4\right )
11=3+2\left (4\right )
15=3+3\left (4\right )
\cdots
187=3+46\left (4\right )
191=3+47\left (4\right )
195=3+48\left (4\right )
The sum S is made up of 3’s and 4’s. How many 3’s are there?
\left (195-3\right )\div 4+1=48+1
=49
S=49\left (3\right )+4\left (1+2+3+\cdots+46+47+48\right )
=147+4\left [\frac{\left (48+1\right )48}{2}\right ]
=147+4\left (49\right )24
=147+4704
=4851
The value of the expression is
4851^2=23532201

Answer: 23532201.

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About mvtrinh

Retired high school math teacher.
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