Answer
a) $630$
b) $1780$
c) $117648$
Work Step by Step
Use formula: $\Sigma_{k=1}^n k=\dfrac{n(n+1)}{2}$ and $\Sigma_{k=1}^n k^2=\dfrac{n(n+1)(n+2)}{6}$
a) Here, we have $\Sigma_{k=9}^{36} (k)=\Sigma_{j=1}^{28} (j+8)=\Sigma_{j=1}^{28} j+\Sigma_{j=1}^{28} 8=630$
b) Here, we have $\Sigma_{k=3}^{17} (k^2)=\Sigma_{j=1}^{15} (j+2)^2)=\Sigma_{j=1}^{15} j^2+4\Sigma_{j=1}^{15}j+\Sigma_{j=1}^{15} 4=1780$
c) Here, we have $\Sigma_{k=18}^{71} k(k-1)=\Sigma_{j=1}^{54} (j+17)(j+16)=\Sigma_{j=1}^{54} j^2+33 \Sigma_{j=1}^{54} j+\Sigma_{j=1}^{54} 272=117648$
