Geometric Progression (GP) and its formula

Geometric Progression (GP) is a sequence of numbers in which each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The formula for the nth term of a GP is given by:

an = a1 * r^(n-1)

where
an = nth term
a1 = first term
r = common ratio
n = number of terms

Solution

Given GP is: 5, 20, 80, 320, ......., 20480

To find the number of terms in the GP, we need to first find the common ratio and the value of n.

Finding the common ratio

We can find the common ratio by dividing any term by the previous term:

20/5 = 4
80/20 = 4
320/80 = 4
20480/5120 = 4

Hence, the common ratio is 4.

Finding the value of n

We know that the nth term of the GP is given by:

an = a1 * r^(n-1)

Substituting the values, we get:

20480 = 5 * 4^(n-1)

Dividing both sides by 5, we get:

4096 = 4^(n-1)

Taking the logarithm of both sides, we get:

log 4096 = log 4^(n-1)

12 = (n-1) * log 4

n-1 = 12/log 4

n-1 = 7.24 (approx)

n = 8.24 (approx)

Since the number of terms has to be a whole number, we take the next integer value of n, which is 9.

Therefore, the number of terms in the GP is 9.

Hence, the correct answer is option E (7).