The problem asks us to find the partial sums $S_1, S_2, \dots, S_n$ for the given sequences: (a) $1, 3, 5, 7, 9, 11$ (b) $4, 2, 0, -2, -4$ Here, $S_n$ represents the sum of the first $n$ terms of the sequence.

1. Problem Description

The problem asks us to find the partial sums $S_1, S_2, \dots, S_n$ for the given sequences:

(a) $1, 3, 5, 7, 9, 11$

(b) $4, 2, 0, -2, -4$

Here, $S_n$ represents the sum of the first $n$ terms of the sequence.

2. Solution Steps

(a) For the sequence $1, 3, 5, 7, 9, 11$, we have:

$S_1 = 1$

$S_2 = 1 + 3 = 4$

$S_3 = 1 + 3 + 5 = 9$

$S_4 = 1 + 3 + 5 + 7 = 16$

$S_5 = 1 + 3 + 5 + 7 + 9 = 25$

$S_6 = 1 + 3 + 5 + 7 + 9 + 11 = 36$

(b) For the sequence $4, 2, 0, -2, -4$, we have:

$S_1 = 4$

$S_2 = 4 + 2 = 6$

$S_3 = 4 + 2 + 0 = 6$

$S_4 = 4 + 2 + 0 + (-2) = 4$

$S_5 = 4 + 2 + 0 + (-2) + (-4) = 0$

3. Final Answer

(a) $S_1 = 1, S_2 = 4, S_3 = 9, S_4 = 16, S_5 = 25, S_6 = 36$

(b) $S_1 = 4, S_2 = 6, S_3 = 6, S_4 = 4, S_5 = 0$