36.  In a geometric progression, the 3rd term is 36 and the 5th term is 324. The 7th term of the same progression will be _______. (in integer)

Geometric Progression Problem: Find the 7th Term When the 3rd Term Is 36 and the 5th Term Is 324

Understanding the Geometric Progression Question

This question is based on the concept of a geometric progression (GP). In a geometric progression, every term after the first term is obtained by multiplying the previous term by a fixed number known as the common ratio. The question gives us the 3rd term and the 5th term of the progression, and we need to calculate the 7th term.

The given information is:

3rd term = 36

5th term = 324

We have to determine the 7th term of the same geometric progression.

General Formula for the nth Term of a Geometric Progression

The general formula for finding any term of a geometric progression is:

T_n = arⁿ⁻¹

In this formula, T_n represents the nth term of the geometric progression, a represents the first term, r represents the common ratio, and n represents the position of the required term.

Writing the 3rd Term of the Geometric Progression

For the 3rd term, we substitute n = 3 into the general formula:

T₃ = ar³⁻¹ = ar²

Since the 3rd term is given as 36, we can write:

ar² = 36     …(1)

Writing the 5th Term of the Geometric Progression

For the 5th term, we substitute n = 5 into the general formula:

T₅ = ar⁵⁻¹ = ar⁴

Since the 5th term is given as 324, we can write:

ar⁴ = 324     …(2)

Finding the Square of the Common Ratio

To find the relationship between the given terms, divide equation (2) by equation (1):

ar⁴ / ar² = 324 / 36

The first term a cancels from the numerator and denominator. Using the exponent rule r^m / rⁿ = r^m−n, we get:

r⁴⁻² = 9

Therefore:

r² = 9

This result is especially useful because the 5th term and the 7th term are also separated by two positions. In a geometric progression, moving forward by two terms means multiplying by r².

Calculating the 7th Term of the Geometric Progression

The 7th term can be written as:

T₇ = ar⁶

We already know that:

T₅ = ar⁴ = 324

Therefore, the ratio of the 7th term to the 5th term is:

T₇ / T₅ = ar⁶ / ar⁴ = r²

Since r² = 9, we obtain:

T₇ = T₅ × 9

Substituting the value of the 5th term:

T₇ = 324 × 9

T₇ = 2916

Why the Direct Term-Ratio Method Works

In a geometric progression, the ratio between two terms depends on the difference between their positions. The 3rd term and the 5th term differ by two positions, so their ratio is r². Similarly, the 5th term and the 7th term also differ by two positions, so their ratio must again be r².

Since:

T₅ / T₃ = 324 / 36 = 9

the same multiplier applies when moving from the 5th term to the 7th term:

T₇ / T₅ = 9

Therefore:

T₇ = 324 × 9 = 2916

Final Answer

The 7th term of the given geometric progression is:

2916

Correct Answer: 2916