Question

4.How many terms of AP 17,13,11.. must be added to
get the sum 72? *
6
Ο Ο Ο
8​

Answer 1

Answer:

Number of terms = 12 or 6

Step-by-step explanation:

Given:

• The A.P is 17, 15, 13, 11....
• Sum of terms = 72

To Find:

• The number of terms required for getting the sum of 72.

Solution:

Here first term of the A.P is 17.

Common difference = 15 - 17

Common difference = -2

Now sum of n terms of an A.P is given by,

Sₙ = n/2 (2a₁ + (n - 1) × d)

where n =  number of terms

           a₁ = first term

           d = common difference

Substituting the data,

72 = n/2 ( 2 × 17 + (n - 1) × -2)

72 × 2 = n ( 34 - 2n + 2)

144 = n( 36 - 2n)

144 = 36 n - 2n²

Divide the whole equation by 2

n² - 18n + 72 = 0

Solving the quadratic equation by splitting the middle term,

n -12n - 6n + 72 = 0

n (n - 12) - 6 (n - 12) = 0

Taking n - 12 as common,

(n - 12) (n - 6) = 0

Either n = 12 or n = 6

Hence the number of terms that must be added to get the sum of 72 can be either 12 or 6.

Verification:

If number of terms = 6,

S₆ = 6/2 ( 2a₁ + (6 - 1) × d)

S₆ = 3 ( 34 + 5 × -2)

S₆ = 3 × 24

S₆ = 72

If number of terms = 12,

S₁₂ = 12/2 (2a1 + (12 - 1) × d)

S₁₂ = 6 ( 34 + 11 × -2)

S₁₂ = 6 × 12

S₁₂ = 72

Hence verified.