Question

Partial Sums Find the first six partial sums $S_1, S_2, S_3, S_4, S_5, S_6$ of the sequence whose nth term is given.

$-1, 1, -1, 1,$….

Short answer

The partial sums are:

$\begin{array}{l}\text{First\hspace{0.17em}\hspace{0.17em}partial\hspace{0.17em}\hspace{0.17em}sum}=S_1=-1.\\ \text{Second\hspace{0.17em}\hspace{0.17em}partial\hspace{0.17em}\hspace{0.17em}sum}=S_2=0.\\ \text{Third\hspace{0.17em}\hspace{0.17em}partial\hspace{0.17em}\hspace{0.17em}sum}=S_3=-1.\\ \text{Fourth\hspace{0.17em}\hspace{0.17em}partial\hspace{0.17em}\hspace{0.17em}sum}=S_4=0\\ \text{Fifth\hspace{0.17em}\hspace{0.17em}partial\hspace{0.17em}\hspace{0.17em}sum}=S_5=-1.\\ \text{Sixth\hspace{0.17em}partial\hspace{0.17em}\hspace{0.17em}sum}=S_6=0.\end{array}$

Step-by-step solution

Step 1.  Given

The given sequence is $-1, 1, -1, 1,$…

Step 2.  To determine

We have to find the first six partial sums.

Step 3.  Calculation

The given sequence is $-1, 1, -1, 1,$…

We see that the terms are alternating signs with 1.

So, the first 6 terms of these sequence are $-1, 1, -1, 1, -1, 1$.

Hence,

$\begin{array}{l}\text{First\hspace{0.17em}\hspace{0.17em}partial\hspace{0.17em}\hspace{0.17em}sum}=S_1=-1.\\ \text{Second\hspace{0.17em}\hspace{0.17em}partial\hspace{0.17em}\hspace{0.17em}sum}=S_2=-1+1=0.\\ \text{Third\hspace{0.17em}\hspace{0.17em}partial\hspace{0.17em}\hspace{0.17em}sum}=S_3=-1+1-1=-1.\\ \text{Fourth\hspace{0.17em}\hspace{0.17em}partial\hspace{0.17em}\hspace{0.17em}sum}=S_4=-1+1-1+1=0.\\ \text{Fifth\hspace{0.17em}\hspace{0.17em}partial\hspace{0.17em}\hspace{0.17em}sum}=S_5=-1+1-1+1-1=-1.\\ \text{Sixth\hspace{0.17em}partial\hspace{0.17em}\hspace{0.17em}sum}=S_6=-1+1-1+1-1+1=0.\end{array}$