Question

The second and fifth term of a geometric progression are 1 and⅛ respectively find the right term

Answer 1

Let the first term of the geometric progression be a and the common ratio be r. Then we have:

Second term = ar = 1
Fifth term = ar^4 = 1/8

Dividing the second equation by the first, we get:

ar^3 = (1/8)/(1) = 1/8

Multiplying the second equation by r^3, we get:

ar^3 = a*r^4 = (1/8)*r^3

Substituting the value of ar^3 from the first equation into the second equation, we get:

1/8 = (1/8)*r^3

Simplifying, we get:

r^3 = 1

Taking the cube root of both sides, we get:

r = 1

Substituting this value of r into the first equation, we get:

a*1 = 1

So, the first term of the geometric progression is 1.

Now, we need to find the right term. Let n be the number of terms in the progression.

The nth term of a geometric progression with first term a and common ratio r is given by:

an = a*r^(n-1)

We know that the second term is 1, so:

a*r = 1

We also know that the fifth term is 1/8, so:

a*r^4 = 1/8

Dividing the second equation by the first, we get:

r^3 = 1/8

Taking the cube root of both sides, we get:

r = 1/2

Substituting this value of r into the first equation, we get:

a*(1/2) = 1

a = 2

Now, we can find the right term using the formula:

an = a*r^(n-1)

We know that the value of r is 1/2 and we need to find the value of n such that an = 2^n is the right term. So, we have:

2^n = 2*(1/2)^(n-1)

2^(n+1) = 1

n+1 = 0 (since 2^(n+1) cannot be 1 for any positive integer n)

n = -1

Since n must be a positive integer, there is no right term in this case.

Therefore, there is no solution to this problem.