Chapter 8, Sequences and Series - Section 8.1 - Sequences and Summation Notation - 8.1 Exercises - Page 601: 40

$S_{1}=1$ $S_{2}=5$ $S_{3}=14$ $S_{4}=30$ $S_{5}=55$ $S_{6}=91$

We are given: $1^2, 2^2, 3^2, 4^2$ ... We notice that the pattern is the square of the next integer: $a_1=1^{2}$ $a_2=2^{2}$ $a_3=3^{2}$ $a_4=4^{2}$ Therefore: $a_{n}=n^{2}$ So: $a_5=5^{2}$ $a_6=6^{2}$ We find the partial sums: $S_{1}=1^{2}=1$ $S_{2}=1+2^{2}=5$ $S_{3}=5+3^{2}=5+9=14$ $S_{4}=14+4^{2}=14+16=30$ $S_{5}=30+5^{2}=30+25=55$ $S_{6}=55+6^{2}=55+36=91$