Question

Use sigma notation to represent the sum of the first six terms of the following sequence: −10, −13, −16, …

Answer 1

Sigma notation of sequence −10, −13, −16, … is $\sum_{n=1}^{6}-(3 n+7)$

Solution:

Need to determine the sigma notation for the following sequence  

−10, −13, −16, …

Let's try to build a generic formula for given sequence  

The given sequence is in Arithmetic progression where first term = -10 and common difference = -3

The formula for arithmetic progression is given as:

$a_n = a_1 + (n - 1)d$

Where,

$a_n$ is the nth term in the sequence

$a_1$ is the first term in the sequence

d is the common difference between the terms

Here in this sequence $a_1$ = -10 and d = -3

$a_n = -10 + (n - 1)(-3)\\\\a_n = -10 - 3n + 3\\\\a_n = -3n - 7$

So generic formula for a term is – ( 3n + 7 )

$\begin{array}{l}{\text { For } \mathrm{n}=1, \text { term is }-(3 \times 1+7)=-10} \\\\ {\text { For } \mathrm{n}=2, \text { term is }-(3 \times 2+7)=-13} \\\\ {\text { For } \mathrm{n}=3, \text { term is }-(3 \times 3+7)=-16}\end{array}$

And so on

Using sigma notation for arithmetic sequence:

$\sum_{k=1}^{n} a_{k}$

So for first six terms value of n will vary from 1 to 6 and in sigma notation it can be represented as

$\sum_{n=1}^{6}-(3 n+7)$