Question

In an A.P., if the first term is 22, the common difference is −4 and the sum to n terms is 64, find n.

Answer 1

Answer:

n is 4 or 8 .

Step-by-step explanation:

Given :  

First term, a = 22, common difference, d = - 4,  sum to n terms, Sn = 64

By using the formula ,Sum of nth terms , Sn = n/2 [2a + (n – 1) d]

64 = n/2 [2 × 22 + (n – 1) × -4]

64 × 2 = n[44 - 4n + 4]

128 = n[48 - 4n]

48n – 4n² = 128

48n – 4n² - 128 = 0

4(12n - n² - 32) = 0

n² – 12n + 32 = 0

n² – 8n – 4n + 32 = 0

[By middle term splitting]

n(n - 8) - 4(n - 8) = 0

(n – 8) (n – 4) = 0

n = 8 or n = 4

Hence n is 4 or 8 .

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Answer 2

Given:-

First term (a) = 22

common difference (d) = -4

Sum of n th terms is 64

To find :-

The value of n.

Solution:-

Let a be the first term and d be the common difference.

According to question:-

a = 22 d = -4

By using the formula of Sn.

$\boxed{\sf{ </strong><strong>sn = \frac{n}{2} </strong><strong>[</strong><strong>2a + (n - 1)d</strong><strong> </strong><strong>]</strong><strong>}}$

by putting the given value,

64 = n /2 [ 2 × 22 + ( n - 1) -4 ]

64 = n/2 [ 44 -4n +4 ]

64 = n/2 [ 48 - 4n]

128 = 48n - 4n²

dividing both sides by 4 we get,

32 = 12n - n²

n² -12n +32 = 0

Now, use middle Splitting term,

n² -8n -4n +32 = 0

n ( n - 8) -4 (n - 8 ) = 0

(n - 4) = 0 or, (n - 8 ) = 0

n = 4 or, n = 8

hence, there is two possible value of n is 4 and 8.