Compute the following two sums: (a) #S_a =Sigma_(k=1)^20(2 − 3k + 2k^2),# (b)#S_b =Sigma_(k=10)^50k#?

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Answer 1 · 2016-06-01T03:27:30

We will use the following formulae:

#sum_(k=1)^n cX_k=csum_(k=1)^nX_k#
#sum_(k=1)^n(X_k+-Y_k)=sum_(k=1)^nX_k+-sum_(k=1)^nY_k#
#sum_(k=1)^n c=cn#
#sum_(k=1)^n k=(n(n+1))/2#
#sum_(k=1)^nk^2=(n(n+1)(2n+1))/6#

(a)

#sum_(k=1)^20(2-3k+2k^2)=sum_(k=1)^20 2-3sum_(k=1)^20 k+2sum_(k=1)^20k^2#

Here, #20=n# in all instances:

#=2*20-3(((20)(20+1))/2)+2((20(20+1)(2*20+1))/6)#

#=40-630+5740=5150#

(b)

Note that:

#overbrace(sum_(k=10)^50k)^("10,11...49,50")=overbrace(sum_(k=1)^50k)^("1,2...49,50")-overbrace(sum_(k=1)^9k)^("1,2...8,9")#

So, this then equals

#((50)(50+1))/2-((9)(9+1))/2=1275-45=1230#