Question

write the series using summation notation. then find the sum of the series.
9 + 16 + 25 + ... +100

∑ from i=1 to □ of □

the sum is □.

Explanation:

Step1: Identify the sequence pattern

Notice that $9=3^2$, $16=4^2$, $25=5^2$, ..., $100=10^2$. The terms are squares of integers starting at 3 and ending at 10. To index from $i=1$, adjust the term: when $i=1$, we need $3^2=(1+2)^2$, so the general term is $(i+2)^2$. The number of terms is $10-3+1=8$, so the upper limit is 8.
Summation notation: $\sum_{i=1}^{8} (i+2)^2$

Step2: Expand the general term

Expand $(i+2)^2$ using the square of a binomial:
$(i+2)^2 = i^2 + 4i + 4$
So the sum becomes $\sum_{i=1}^{8} (i^2 + 4i + 4) = \sum_{i=1}^{8}i^2 + 4\sum_{i=1}^{8}i + \sum_{i=1}^{8}4$

Step3: Apply summation formulas

Use the formulas:

• $\sum_{i=1}^{n}i^2 = \frac{n(n+1)(2n+1)}{6}$
• $\sum_{i=1}^{n}i = \frac{n(n+1)}{2}$
• $\sum_{i=1}^{n}c = nc$ (where $c$ is a constant)

For $n=8$:

1. $\sum_{i=1}^{8}i^2 = \frac{8(8+1)(2*8+1)}{6} = \frac{8*9*17}{6} = 204$
2. $4\sum_{i=1}^{8}i = 4*\frac{8(8+1)}{2} = 4*\frac{8*9}{2} = 4*36 = 144$
3. $\sum_{i=1}^{8}4 = 8*4 = 32$

Step4: Sum the results

Add the three computed values together:
$204 + 144 + 32$

Answer:

Summation notation: $\sum_{i=1}^{8} (i+2)^2$
The sum is $380$