5

th term & 3rd term of GP are 256 & 16 respectively. Find first term, common ratio of GP. Also

find 7th term & sum of first 5 terms of the series

To solve this problem, let's denote the geometric progression (GP) as \( a, ar, ar^2, ar^3, ar^4, \ldots \), where:
- \( a \) is the first term, and
- \( r \) is the common ratio.

### Given:
- The 3rd term is 16: \( ar^2 = 16 \)
- The 5th term is 256: \( ar^4 = 256 \)

### Step 1: Find the common ratio \( r \)

Dividing the equation for the 5th term by the equation for the 3rd term, we get:
\[
\frac{ar^4}{ar^2} = \frac{256}{16}
\]
\[
r^2 = 16
\]
\[
r = \pm 4
\]

### Step 2: Find the first term \( a \)

Using \( r = 4 \), substitute back into the equation for the 3rd term:
\[
a \cdot 4^2 = 16
\]
\[
16a = 16
\]
\[
a = 1
\]

Alternatively, using \( r = -4 \):
\[
a \cdot (-4)^2 = 16
\]
\[
16a = 16
\]
\[
a = 1
\]

So, the first term \( a = 1 \) for both cases.

### Step 3: Find the 7th term of the series

The 7th term is given by:
\[
a r^6 = 1 \cdot (\pm 4)^6 = 1 \cdot 4096 = 4096
\]

### Step 4: Find the sum of the first 5 terms

The sum of the first \( n \) terms of a geometric progression is given by:
\[
S_n = a \frac{r^n - 1}{r - 1}, \text{ for } r \neq 1
\]
Substitute \( a = 1 \), \( r = 4 \), and \( n = 5 \):
\[
S_5 = 1 \cdot \frac{4^5 - 1}{4 - 1} = \frac{1024 - 1}{3} = \frac{1023}{3} = 341
\]

For \( r = -4 \):
\[
S_5 = 1 \cdot \frac{(-4)^5 - 1}{-4 - 1} = \frac{-1024 - 1}{-5} = \frac{-1025}{-5} = 205
\]

### Summary
- First term, \( a \) = 1
- Common ratio, \( r \) = 4 or -4
- 7th term = 4096
- Sum of the first 5 terms = 341 (for \( r = 4 \)) or 205 (for \( r = -4 \))

Would you like to go through any step in more detail, or have any questions?

### Related Questions:
1. How do you determine the common ratio in a geometric progression?
2. What happens to a geometric progression if the common ratio is a fraction?
3. How does the formula for the sum of a geometric series change when \( r = 1 \)?
4. What is the effect of a negative common ratio on the terms of a geometric sequence?
5. How do you find the sum to infinity of a geometric series?

**Tip:** The sum of an infinite geometric series converges only if the common ratio \( |r| < 1 \).

5th term & 3rd term of GP are 256 & 16 respectively. Find first term, common ratio of GP. Also find 7th term & sum of first 5 terms of the series.

Solve a geometric progression problem where the 5th and 3rd terms are given as 256 and 16, respectively. Find the first term, common ratio, the 7th term, and the sum of the first 5 terms using key GP formulas. This problem is ideal for students in Grades 9-11.

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