Question

For a sequence if s_(n)=5(4^(n)-1) find the nth term hence verify it is G.P.also find r ​

Answer 1

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The 9th term of the sequence is 6.

Step-by-step explanation:

Given : A sequence S_n=2(3n-1)Sn=2(3n−1)

To find : The nth term and hence show that the sequence is a G.P?

Solution :

The sum of sequence is S_n=2(3n-1)=6n-2Sn=2(3n−1)=6n−2

a_n=S_n-S_{n-1}an=Sn−Sn−1

a_n=6n-2-(6(n-1)-2)an=6n−2−(6(n−1)−2)

a_n=6n-2-(6n-6-2)an=6n−2−(6n−6−2)

a_n=6n-2-(6n-8)an=6n−2−(6n−8)

a_n=6n-2-6n+8an=6n−2−6n+8

a_n=6an=6

The nth term of the sequence is 6.

Substitute n=1 in sum of sequence,

S_1=2(3(1)-1)S1=2(3(1)−1)

S_1=2(2)S1=2(2)

S_1=4S1=4

The first term of sequence is a=4

Substitute n=2 in sum of sequence,

S_2=2(3(2)-1)S2=2(3(2)−1)

S_2=2(5)S2=2(5)

S_2=10S2=10

i.e. Sum of first two term is 10.

a_1+a_2=10a1+a2=10

4+a_2=104+a2=10

a_2=6a2=6

nth term is also 6.

Which means the sequence has only 2 terms

The first term is a=4 and second term is 6.