Determine the third term of a geometrical progression whose first and second term are 2 and 54 respectively

1. Analyze the question

We are given a Geometric Progression (GP) where:

• The first term ($a$) = 2
• The second term ($ar$) = 54
• We need to find the third term ($ar^2$).

2. Explain the concept

In a Geometric Progression, each term is obtained by multiplying the previous term by a constant called the common ratio ($r$).

• The first term is $a$.
• The second term is $ar$.
• The third term is $ar^2$.

To find the third term, we first need to find the common ratio ($r$) by dividing the second term by the first term.

3. Show the solution

Step 1: Find the common ratio ($r$) Since the second term is $ar = 54$ and the first term $a = 2$:

Step 2: Find the third term ($ar^2$) The third term is the second term multiplied by the common ratio ($r$):

Let's calculate $54 \times 27$: $54 \times 20 = 1080$ $54 \times 7 = 378$ $1080 + 378 = 1458$

4. Identify the answer

Comparing our result to the given options: A. 1458 B. 1485 C. 1345 D. 1258

The correct option is A.

5. Tip

In a GP, remember the relationship: (Second term)² = (First term) × (Third term). You can use this to verify your answer: $54^2 = 2916$, and $2 \times 1458 = 2916$. Since they match, you can be confident in your result!