Question

4. A sequence (an) is given by the formula an = 10 – 3n, is an AP, find the common difference of the sequence.​

Answer 1

Provided that: A sequence (AP) is given by the formula ${\sf{a_n \: = 10 - 3n}}$ , is it an AP!? Find the common difference of the sequence.

Knowledge required:

→ d = 2nd term - 1st term

→ d = 4th term - 3rd term

Required solution:

~ Firstly let us put n as 1

${\sf{:\implies a_n \: = 10 - 3n}}$

${\sf{:\implies a_1 \: = 10 - 3(1)}}$

${\sf{:\implies a_1 \: = 10 - 3}}$

${\pmb{\sf{:\implies a_1 \: = 7}}}$

~ Now let us put n as 2

${\sf{:\implies a_n \: = 10 - 3n}}$

${\sf{:\implies a_2 \: = 10 - 3(2)}}$

${\sf{:\implies a_2 \: = 10 - 6}}$

${\pmb{\sf{:\implies a_2 \: = 4}}}$

~ Now let's put n as 3

${\sf{:\implies a_n \: = 10 - 3n}}$

${\sf{:\implies a_3 \: = 10 - 3(3)}}$

${\sf{:\implies a_3 \: = 10 - 9}}$

${\pmb{\sf{:\implies a_3 \: = 1}}}$

~ Now let us put n as 4

${\sf{:\implies a_4 \: = 10 - 3n}}$

${\sf{:\implies a_4 \: = 10 - 3(4)}}$

${\sf{:\implies a_4 \: = 10 - 12}}$

${\pmb{\sf{:\implies a_4 \: = -2}}}$

Therefore, series be like 7, 4, 1, -2...

~ Now let us find out the common difference by using suitable formula!

→ d = 2nd term - 1st term

→ d = 4 - 7

→ d = -3

___________________________

→ d = 4th term - 3rd term

→ d = -2 - 1

→ d = -3

• Henceforth, d are same in both the conditions, therefore, the series in an arithmetic series. And here the common difference is -3