Question

Write the sum using summation notation, assuming the suggested pattern continues.

Answer 1

The numbers increase by 3 starting at 0 use the equation for arithmetic sequence.

n
∑ 3x-1
i = 0


proper notion is shown in the picture.

Answer 2

Answer:

345

Step-by-step explanation:

We know that $1+2+3+...+n=\frac{n(n+1)}{2}$

Given pattern : $-1 + 2 + 5 + 8 + ... + 44$

-1 + 2 + 5 + 8 + ... + 44 = \sum_{i=0}^{15} 3x-1

We can write $\sum_{i=0}^{15} 3x-1=3\sum_{i=0}^{15}x-\sum_{i=0}^{15}1$

Here, $3\sum_{i=0}^{15}x=3\left ( 1+2+3+...+15 \right )$

On putting n = 15 in $\frac{15(15+1)}{2}=\frac{15\times 16}{2}=120$

So, $3\sum_{i=0}^{15}x=3(120)=360$

Also, $\sum_{i=0}^{15}1=15$

Therefore,

$-1 + 2 + 5 + 8 + ... + 44\\=\sum_{i=0}^{15} \left ( 3x-1 \right )\\=3\sum_{i=0}^{15}x-\sum_{i=0}^{15}1\\=3\left ( 0+1+2+3+...+15 \right )-15\\=3\left [ \frac{15\left ( 15+1 \right )}{2} \right ]-15\\=360-15\\=345$