The series' value is maximized if the sum only consists of positive terms. Notice that each term in the sum takes the form 4<em>n</em> + 1 for integer <em>n</em>. This means the smallest positive integer that the sum can involve is 1, so the maximum value is
<em>S</em> = 29 + 25 + 21 + … + 9 + 5 + 1
Reversing the order of terms gives the same sum,
<em>S</em> = 1 + 5 + 9 + … + 21 + 25 + 29
Adding terms in the same positions gives us twice this sum,
2<em>S</em> = (29 + 1) + (25 + 5) + (21 + 9) + … + (1 + 29)
Notice how each grouped sum adds to 30. There are 8 terms in the sum, since 4<em>n</em> + 1 = 29 when <em>n</em> = 8. So
2<em>S</em> = 8 × 30 = 240 ===> <em>S</em> = 120
