Problem:
What is the value of
23−13+43−33+63−53+⋯+183−173?
Answer Choices:
A. 2023
B. 2679
C. 2941
D. 3159
E. 3235
Solution:
The formula for the sum of the first n cubes is useful here:
13+23+33+⋯+n3=(2n(n+1))2
Therefore
23+43+63+⋯+183=8(13+23+33+⋯+93)=8⋅452=16,200
Also,
13+33+53+⋯+173=(13+23+33+⋯+173)−(23+43+63+⋯+163)=(17⋅9)2−8⋅362=23,409−10,368=13,041
The requested value is 16,200−13,041=(D)3159.
OR
Recall the formulas for the sum of the first n positive integers and for the sum of the first n squares:
1+2+3+⋯+n=2n(n+1)
and
12+22+32+⋯+n2=6n(n+1)(2n+1)
In summation notation the given expression is
n=1∑9((2n)3−(2n−1)3)=n=1∑9(8n3−(8n3−12n2+6n−1))=n=1∑9(12n2−6n+1)=12⋅69⋅10⋅19−6⋅29⋅10+9=3420−270+9=(D)3159
The problems on this page are the property of the MAA's American Mathematics Competitions