Home » Blog » Establish a formula for alternating sum of squares

# Establish a formula for alternating sum of squares

Claim: Proof. For , we have 1 on the left, and on the right . Thus, the formula is true for .
Assume then that it is true for some . Then, Thus, if the formula is true for then it is true for . Since we established it is true for , we have that it is true for all 1. luis says:

I don’t understand the 3rd step, please somebody explain me. Thanks.

• The numbers in the series alternate between negative and positive. So if you pick one number from the series and it’s negative, the next number will be positive.

• Tiago says:

Hi Luis. He used the identity: (1+2+…+k) = k(k+1)/2 (this appears in part (a) of the exercise); and he put the factor (-1)^(k+2) out. Notice that you have [(-1)^(k+1)] * [k(k+1)/2], so when you put (-1)^(k+2) out, what stays inside the parenthesis is: [(-1)^(-1)] * [k(k+1)/2]. But (-1)^(-1) = -1. Hence you get, – k(k+1)/2.

Remark: remember that (a^b) * (a^c) = a^(b+c).