Question

the arithmetic sequence ai is defined by the formula: a1= 2 ai= ai - 1 -3 find the sum of the first 335 terms in the sequence

Answer 1

Given:

In an arithmetic sequence,

$a_1=2$

$a_i=a_{i-1}-3$

To find:

The sum of the first 335 terms in the given sequence.

Solution:

The recursive formula of an arithmetic sequence is:

$a_i=a_{i-1}+d$         ...(i)

Where, d is the common difference.

We have,

$a_i=a_{i-1}-3$          ...(ii)

On comparing (i) and (ii), we get

$d=-3$

The sum of first i terms of an arithmetic sequence is:

$S_i=\dfrac{i}{2}[2a+(i-1)d]$

Putting $i=335,a=2,d=-3$, we get

$S_{335}=\dfrac{335}{2}[2(2)+(335-1)(-3)]$

$S_{335}=\dfrac{335}{2}[4+(334)(-3)]$

$S_{335}=\dfrac{335}{2}[4-1002]$

$S_{335}=\dfrac{335}{2}(-998)$

On further simplification, we get

$S_{335}=335\times (-499)$

$S_{335}=-167165$

Therefore, the sum of the first 335 terms in the given sequence is -167165.