Geometric Progression

We are given that the first term of the geometric progression (G.P.) is 3 and the common ratio is 2. We are asked to find the sum of the first eight terms of the G.P.

The formula for the sum of the first terms of a G.P. is:

where is the first term, is the common ratio, and is the number of terms.

We are given:

Substituting these values into the formula:

The sum of the first eight terms of the G.P. is 765.

Given:

• The first term of the geometric progression (G.P.) is 1.
• The sum of the 3rd and 5th terms is 90.

Formula for nth term of a G.P.:

The nth term of a geometric progression is given by:

T_n = a * r^n-1

where a is the first term and r is the common ratio.

Step 1: Express the 3rd and 5th terms of the G.P.

We know the first term (a) is 1, so:

3rd term (T₃) = 1 * r^3-1 = r²

5th term (T₅) = 1 * r^5-1 = r⁴

Step 2: Use the given sum of the 3rd and 5th terms.

The sum of the 3rd and 5th terms is 90:

r² + r⁴ = 90

Step 3: Solve for r.

Factor out r² from the left side:

r²(1 + r²) = 90

Now, let x = r². The equation becomes:

x(1 + x) = 90

Expanding:

x + x² = 90

x² + x - 90 = 0

This is a quadratic equation. We can solve it using the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

For the equation x² + x - 90 = 0, a = 1, b = 1, and c = -90. Substituting these values into the quadratic formula:

x = (-1 ± √(1² - 4(1)(-90))) / 2(1)

x = (-1 ± √(1 + 360)) / 2

x = (-1 ± √361) / 2

x = (-1 ± 19) / 2

x = 9 or x = -10

Step 4: Find r.

Since x = r², we have:

r² = 9, so r = 3 (positive root).

Answer:

The positive common ratio of the G.P. is 3.

Let the first term of the geometric progression (G.P.) be and the common ratio be . The second term is , and the third term is . We are given that the sum of the second and third terms is 60, i.e., Factor out the common factor 3: Rearrange the equation: Now solve this quadratic equation using the quadratic formula: Thus, The two possible values for are:

The correct option is (D) : or

-------(1)

--------(2)

Now comparing from above cases (1) and (2)we get:

on solving we get :

Therefore from parent equation we get

Then, sequence becomes

Therefore ,

We are given a geometric progression (GP) sequence: .

Step 1: Understand the GP structure

In a GP, each term is obtained by multiplying the previous term by a common ratio . Therefore:

Step 2: Solve for the common ratio

From the last equation:

Step 3: Find and

Using :

Step 4: Verify the sequence

The sequence becomes , which is indeed a GP with ratio 3.

Step 5: Calculate

Alternative Approach:

Note that could also be negative ( ):

However, since and must be between 3 and 81, we discard .

Final Answer:

The value of is .