Question

The sum of 3 term of an AP is 15 and their product is 80 find the AP

Answer 1

Let the three terms of the arithmetic progression (AP) be \( a - d, a, a + d \), where \( a \) is the middle term and \( d \) is the common difference.

1. The sum of the three terms is given as:
\[
(a - d) + a + (a + d) = 3a
\]
Setting this equal to 15, we have:
\[
3a = 15 \implies a = 5
\]

2. The product of the three terms is given as:
\[
(a - d) \cdot a \cdot (a + d) = 80
\]
Using the value of \( a \):
\[
(5 - d) \cdot 5 \cdot (5 + d) = 80
\]

3. Simplifying the product:
\[
5 (5 - d)(5 + d) = 80
\]
Using the difference of squares:
\[
(5 - d)(5 + d) = 25 - d^2
\]
Thus:
\[
5(25 - d^2) = 80
\]
Dividing by 5:
\[
25 - d^2 = 16
\]
Rearranging gives:
\[
d^2 = 25 - 16 = 9
\]
Taking the square root:
\[
d = 3 \quad \text{or} \quad d = -3
\]

4. Now we can find the three terms of the AP:

If \( d = 3 \):
\[
5 - 3 = 2, \quad 5, \quad 5 + 3 = 8
\]
So, the AP is \( 2, 5, 8 \).

If \( d = -3 \):
\[
5 - (-3) = 8, \quad 5, \quad 5 + (-3) = 2
\]
The AP would still be the same set of numbers \( 8, 5, 2 \), just in a different order.

Thus, the arithmetic progression is \( 2, 5, 8 \).