# The sum of the third and the seventh terms of an AP is 6 and their product is 8. Find the sum of first sixteen terms of the AP

**Solution:**

A sequence that has a common difference between any pair of consecutive numbers is called an Arithmetic Progression.

n^{th} term of an AP is aₙ = a + (n - 1)d

Here, a is the first term, d is the common difference and n is the number of terms.

Given:

- a₃ + a₇ = 6 ----- (1)
- a₃ × a₇ = 8 ----- (2)

We know that n^{th} term of AP is aₙ = a + (n - 1)d

Third term, a₃ = a + (3 - 1)d

a₃ = a + 2d ----- (3)

Seventh term,

a₇ = a + (7 - 1)d

a₇ = a + 6d ----- (4)

Using equation (3) and equation (4) in equation (1) to find the sum of the terms,

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

2a + 8d = 6

a + 4d = 3

a = 3 - 4d ----- (5)

Using equation (3) and equation (4) in equation (2) to find the product of the terms,

(a + 2d ) × (a + 6d ) = 8

Substituting the value of a from equation (5) above,

(3 - 4d + 2d) × (3 - 4d + 6d) = 8

(3 - 2d) × (3 + 2d) = 8

(3)² - (2d)² = 8 [Since (a + b)(a - b) = a² - b² ]

9 - 4d² = 8

4d² = 1

d² = 1/4

d = ½, -½

Case 1: When d = ½

a = 3 - 4d

= 3 - 4 × ½

= 3 - 2

= 1

Sₙ = n/2 [2a + (n - 1) d]

S₁₆ = 16 / 2 [ 2 × 1 + (16 - 1) × ½ ]

= 8 × 19/2

= 76

Case 2: When d = - ½

a = 3 - 4d

= 3 - 4 × (- ½)

= 3 + 2

= 5

Sₙ = n/2 [2a + (n - 1) d]

S₁₆ = 16/2 [2 × 5 + (16 - 1) × (- ½)]

= 8 [10 - 15/2]

= 8 × 5/2

= 20

**☛ Check: **NCERT Solutions for Class 10 Maths Chapter 5

**Video Solution:**

## The sum of the third and the seventh terms of an AP is 6 and their product is 8. Find the sum of first sixteen terms of the AP

Class 10 Maths NCERT Solutions Chapter 5 Exercise 5.4 Question 2

**Summary:**

The sum of the third and the seventh terms of an A.P is 6 and their product is 8 then the sum of first 16 terms of the A.P. is equal to 20 or 76.

**☛ Related Questions:**

- Which term of the A.P : 121, 117, 113, ..., is its first negative term?[Hint : Find n for an less than o]
- A ladder has rungs 25 cm apart. (see Fig. 5.7). The rungs decrease uniformly in length from 45 cm at the bottom to 25 cm at the top. If the top and bottom rungs are 2 ½ m apart, what is the length of the wood required for the rungs? [Hint: number of rungs = 250 / 25 + 1]
- The houses of a row are numbered consecutively from 1 to 49. Show that there is a value of x such that the sum of the numbers of the houses preceding the house numbered x is equal to the sum of the numbers of the houses following it. Find this value of x. [Hint Sx - 1 = S49 - Sx ]
- A small terrace at a football ground comprises of 15 steps each of which is 50 m long and built of solid concrete. Each step has a rise of ¼ m and a tread of ½ m . (see Fig. 5.8). Calculate the total volume of concrete required to build the terrace. [Hint : Volume of concrete required to build the first step =¼ × ½ × 50 m³.].

visual curriculum