Geometric Progression

Geometric Progression (GP) is a sequence of numbers in which each term after the first is obtained by multiplying the preceding term by a fixed, non-zero number called the common ratio (r).

In this problem, we are given a GP with the first term (go) as 5, 10, 20, 40, 80. We need to find the 9th term, 47th term, and the general nth term of this GP.

Finding the 9th Term

To find the 9th term of the GP, we need to determine the common ratio (r). We can do this by dividing any term by its preceding term.

Let's take the second term (10) and divide it by the first term (5):

r = 10/5 = 2

Now, we can use the formula to find the nth term of a GP:

Tn = go * r^(n-1)

Plugging in the values, we get:

T9 = 5 * 2^(9-1)
= 5 * 2^8
= 5 * 256
= 1280

Therefore, the 9th term of the GP is 1280.

Finding the 47th Term

Using the same approach, we can find the common ratio (r) by dividing any term by its preceding term. Let's take the third term (20) and divide it by the second term (10):

r = 20/10 = 2

Now, we can find the 47th term using the formula:

T47 = 5 * 2^(47-1)
= 5 * 2^46

Calculating this value might result in a very large number, which may not be practical to write here. However, you can use a calculator or a computer program to find the exact value.

Finding the nth Term

To find the general nth term of the GP, we can use the formula:

Tn = go * r^(n-1)

Therefore, the nth term of the GP is given by multiplying the first term (go) by the common ratio (r) raised to the power of (n-1).

In this case, go = 5 and r = 2.

Tn = 5 * 2^(n-1)

In summary:
- The 9th term of the GP is 1280.
- The 47th term can be calculated using the formula T47 = 5 * 2^46.
- The general nth term of the GP is Tn = 5 * 2^(n-1).