In this post, we will walk through the step-by-step process of solving a problem involving an arithmetic progression (AP). The sum of the third and seventh terms of an AP is given as 6, and their product is given as 8. Our goal is to find the sum of the first sixteen terms of the AP. Along the way, we will break down each step so you can fully understand how to approach similar problems in the future.

What is an Arithmetic Progression (AP)?

An arithmetic progression is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference (d). The general formula for the nth term of an AP is:

a_n = a + (n – 1) d

Step 1: Define the Given Conditions of Arithmetic progression

We know that:

The 3rd term of the AP is given by:

a₃ = a + 2d

a₇ = a + 6d

We are also told that:

a₃ + a₇ = 6

a₃ X a₇ = 8

Step 2: Solving the Sum of Terms of arithmetic progression

Thus, using the formula for the 3rd and 7th terms, we can substitute them into the first condition:

(a + 2d) + (a + 6d) = 6

2a + 8d = 6

a + 4d = 3

a = 3 – 4d

Step 3: Solving the Product of Terms

Thus, let’s use the second condition, which is the product of the third and seventh terms:

(a + 2d) (a + 6d) = 8

Substitute a = 3-4d

(3 – 4d + 2d) (3 – 4d + 6d) = 8

(3 – 2d) (3 + 2d) = 8

9 – 4d² = 8

4d² = 1

d² = 1/4

d = +or – 1/2

Step 4: Finding the First Term

Thus, that we know  or let’s substitute these values back into the equation.

For:

a + 4(1/2) = 3

a + 2 = 3

a = 1

For:

a + 4(-1/2) = 3

a – 2 = 3

a = 5

Step 5: Finding the Sum of the First Sixteen Terms

Thus, we have the possible values for and , let’s focus on finding the sum of the first sixteen terms of the AP. The formula for the sum of the first n terms of an AP is:

S_n = (n/2) (2a + (n – 1) d)

Let’s calculate for both cases:

1. When a= 1 and d = (1/2)



S₁₆ = (16/2) (2 x 1 + (16 – 1) (1/2)

8 (2 + 7.5)

8 x 9.5

Thus, S₁₆=76

2. When a=5 and d = (-1/2)



S₁₆ = (16/2) (2 x 5 + (16 – 1) (-1/2)

8(10 – 7.5)

8 x2.5

Thus, S₁₆= 20

Thus, the sum of the first sixteen terms of the AP is either 76 or 20, depending on the values of a and d.