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Evaluate the sums in Exercises 2328.

j=1mk=1n(j-k)

Short Answer

Expert verified

The value of the summation is12mn(m-n)

Step by step solution

01

Step 1. Given information

An expression is given asj=1mk=1n(j-k)

02

Step 2. Evaluating summation

First separate out the term then simplify,

j=1mk=1m(j-k)=j=1mk=1n(j-k)k=1n(j-k)=(j-1)+(j-2)+(j-3)++(j-n)=n×j-(1+2+3++n)=n×j-n(n+1)2

Now simplify the other terms in the same manner,

j=1mk=1n(j-k)=j=1mk=1n(j-k)=j=1mn×j-n(n+1)2=n×1-n(n+1)2+n×2-n(n+1)2+n×3-n(n+1)2+.......+n×m-n(n+1)2=n(1+2+3++m)-m×n(n+1)2=n×m(m+1)2-m×n(n+1)2=mnm+12-n+12=mnm-n2=12mn(m-n)

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