Understanding Geometric Progressions and Finding the Tenth Term

Geometric progressions (or sequences) are sequences of numbers where each term is a constant multiple of the previous one. The general formula for the nth term of a geometric progression (GP) is given by:

gn g1 * r^(n-1)

Where: g1 is the first term of the sequence. r is the common ratio (the factor by which each term is multiplied to get the next term). n is the term number. Given the problem: The first term is 2 and the common ratio is 3, we can derive the formula for the nth term as follows:

Formulating the Formula

The formula for the nth term in this sequence is:

gn 2 * 3^(n-1)

To find the tenth term (g10), we substitute n 10 into the formula:

g10 2 * 3^(10-1)

Let's simplify this:

g10 2 * 3^9

3^9 19683

g10 2 * 19683

g10 39366

Therefore, the tenth term of the geometric sequence is 39366.

Overview of the Geometric Sequence

Geometric sequences can also be visualized or listed for better understanding. For the given sequence with a first term of 2 and a common ratio of 3, the sequence looks like this: 2 2 * 3 6 6 * 3 18 18 * 3 54 54 * 3 162 162 * 3 486 By inspection, the fifth term of this sequence is 162. We can verify this using the general formula:

T5 2 * 3^5 - 1 2 * 3^4 2 * 81 162

Further Analysis: Finding the Ninth Term

To find the ninth term (T9) in a similar manner, we need to know the first term (a) and the common ratio (r). The formula to find the n-th term is:

Tn a * r^(n-1)

For the given values where a 3 and r 2, the calculation is as follows:

T9 3 * 2^(9-1) 3 * 2^8 3 * 256 768

Thus, the ninth term of the sequence is 768. This can also be verified as follows:

T9 3 * 256 768

Common Terms in Geometric Progression

Geometric progressions often have specific terms that need to be found. For instance, the ninth term (T9) can be calculated using the formula:

T9 3 * 256 768

Thus, the ninth term of the given sequence is 768. We can derive this using the general formula for the nth term, which is:

Tn a * r^(n-1)

For a 3 and r 2, we get:

T9 3 * 2^8 3 * 256 768