Finding the 8th Term of a Geometric Progression (G.P.)

A Geometric Progression (G.P.) is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

The given sequence is 3, 6, 12, 24, … This is a Geometric Progression.

Identifying the First Term and Common Ratio

In the given G.P.:

• The first term, denoted by \(a\), is the first number in the sequence. So, \(a = 3\).
• The common ratio, denoted by \(r\), is found by dividing any term by its preceding term.

Let's calculate the common ratio:

• \(r = \frac{\text{2nd term}}{\text{1st term}} = \frac{6}{3} = 2\)
• \(r = \frac{\text{3rd term}}{\text{2nd term}} = \frac{12}{6} = 2\)
• \(r = \frac{\text{4th term}}{\text{3rd term}} = \frac{24}{12} = 2\)

The common ratio \(r\) is consistently 2.

Using the Nth Term Formula for a G.P.

The formula to find the \(n\)-th term of a Geometric Progression is:

\(a_n = ar^{n-1}\)

Where:

• \(a_n\) is the \(n\)-th term
• \(a\) is the first term
• \(r\) is the common ratio
• \(n\) is the term number we want to find

Calculating the 8th Term

We need to find the 8th term of the G.P., so \(n = 8\). We have \(a = 3\) and \(r = 2\).

Substitute these values into the nth term formula:

\(a_8 = a \times r^{8-1}\)

\(a_8 = 3 \times 2^{7}\)

Now, we calculate the value of \(2^{7}\):

\(2^1 = 2\)
\(2^2 = 4\)
\(2^3 = 8\)
\(2^4 = 16\)
\(2^5 = 32\)
\(2^6 = 64\)
\(2^7 = 128\)

So, \(a_8 = 3 \times 128\).

Performing the multiplication:

\(3 \times 128 = 384\)

Therefore, the 8th term of the given Geometric Progression is 384.

This calculation uses the fundamental formula for the nth term of a G.P. sequence, making it straightforward to find any term once the first term and common ratio are known.