If $S = \sum_{n=1}^{10} (2n + \frac{1}{2})$, then S is-

Explanation

$S = \sum_{n=1}^{10} (2n + \frac{1}{2})$ $n = 1, 2, 3, ........................., 10$ $S_1 = 2 \times 1 + \frac{1}{2} = \frac{5}{2}$ $S_2 = 2 \times 2 + \frac{1}{2} = \frac{9}{2}$ $S_3 = 2 \times 3 + \frac{1}{2} = \frac{13}{2}$ $S_{10} = 2 \times 10 + \frac{1}{2} = \frac{41}{2}$ $\therefore S = \frac{5}{2} + \frac{9}{2} + \frac{13}{2} + ......... + \frac{41}{2}$ $= \frac{1}{2} [5 + 9 + 13 + ......... + 41] = [115]$ $= \frac{1}{2} [\frac{10}{2} \{2 \times 5 + (10 - 1)4\}]$ $\therefore S_n = \frac{n}{2} [2a + (n - 1)d]$ $= \frac{1}{2} \times 5 [10 + 9 \times 4]$ $= \frac{1}{2} \times 5 \times 46 = 5 \times 23 = 115$ On putting, x=3 in expression $(3)^3 + 3y - 3 = 88$ $y = \frac{64}{3}$ So, the next value of x will be- $\frac{10}{3} + 4 = \frac{22}{3}, \frac{22}{3} + 6 = \frac{40}{3}$ $\frac{40}{3} + 8 = \frac{64}{3}, \frac{64}{3} + 10 = \frac{94}{3}$ then the fifth value of expression will be- $(4)^3 + 3 \times \frac{94}{3} - 3$ $= 64 + 94 - 3$ $= 64 + 91$ $= 155$ So, option (b) is correct answer