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.

$1^2, 2^2, 3^2, 4^2,$….

Short answer

The partial sums are:

$\begin{array}{l}\text{First\hspace{0.17em}\hspace{0.17em}partial\hspace{0.17em}\hspace{0.17em}sum}=S_1=1.\\ \text{Second\hspace{0.17em}\hspace{0.17em}partial\hspace{0.17em}\hspace{0.17em}sum}=S_2=5.\\ \text{Third\hspace{0.17em}\hspace{0.17em}partial\hspace{0.17em}\hspace{0.17em}sum}=S_3=14.\\ \text{Fourth\hspace{0.17em}\hspace{0.17em}partial\hspace{0.17em}\hspace{0.17em}sum}=S_4=30.\\ \text{Fifth\hspace{0.17em}\hspace{0.17em}partial\hspace{0.17em}\hspace{0.17em}sum}=S_5=55.\\ \text{Sixth\hspace{0.17em}partial\hspace{0.17em}\hspace{0.17em}sum}=S_6=91.\end{array}$

Step-by-step solution

Step 1.  Given

The given sequence is$1^2,2^2,3^2,4^2,$ …

Step 2.  To determine

We have to find the first six partial sums.

Step 3.  Calculation

The given sequence is $1^2, 2^2, 3^2, 4^2,$…

We see that the terms are increasing by 1 and they are being squared.

So, the first 6 terms of these sequence are $1^2, 2^2, 3^2, 4^2, 5^2, 6^2$.

Hence,

$\begin{array}{l}\text{First\hspace{0.17em}\hspace{0.17em}partial\hspace{0.17em}\hspace{0.17em}sum}=S_1=1^2=1.\\ \text{Second\hspace{0.17em}\hspace{0.17em}partial\hspace{0.17em}\hspace{0.17em}sum}=S_2=1^2+2^2=5.\\ \text{Third\hspace{0.17em}\hspace{0.17em}partial\hspace{0.17em}\hspace{0.17em}sum}=S_3=1^2+2^2+3^2=14.\\ \text{Fourth\hspace{0.17em}\hspace{0.17em}partial\hspace{0.17em}\hspace{0.17em}sum}=S_4=1^2+2^2+3^2+4^2=30.\\ \text{Fifth\hspace{0.17em}\hspace{0.17em}partial\hspace{0.17em}\hspace{0.17em}sum}=S_5=1^2+2^2+3^2+4^2+5^2=55.\\ \text{Sixth\hspace{0.17em}partial\hspace{0.17em}\hspace{0.17em}sum}=S_6=1^2+2^2+3^2+4^2+5^2+6^2=91.\end{array}$