Question :
The sum of the third and the seventh terms of an AP is 6 and their product is 8. Find the sum of the first sixteen terms of the AP.
Subject
Mathematics
Standard
Class 11
Views
1911
Asked By
Michael
Qayanat
Answer / Solution
Answer / Solution
Solution:
From the given statements, a3 + a7 = 6 .....(i) and a3 × a7 = 8 ........(ii),
By using the nth term formula, an = a + (n − 1)d, the third term will be
Third term, a3 = a + (3 − 1)d ⇒ a3 = a + 2d ………(iii) and
seventh term, a7 = a + (7 -1)d ⇒ a7 = a + 6d ....... (iv)
Putting in equation (i) from the values from equation (iii) and (iv), we get,
a + 2d + a + 6d ⇒ 6 2a + 8d = 6 ⇒ a + 4d = 3
⇒ a = 3 – 4d ………(v)
Again putting the eq. (iii) and (iv), in eq. (ii), we get, (a + 2d) × (a + 6d) = 8
Putting the value of a from equation (v), we get,
(3 – 4d + 2d) × (3 – 4d + 6d) = 8
(3 – 2d) × (3 + 2d) = 8
32 – 2d2 = 8
9 – 4d2 = 8
4d2 = 1
d = 1/2 or –1/2
Now, by putting both the values of d, we get,
a = 3 – 4d = 3 – 4(1/2) = 3 – 2 = 1, when d = 1/2
a = 3 – 4d = 3 – 4(– 1/2) = 3 + 2 = 5, when d = – 1/2
We know, the sum of nth term of AP is; Sn = n/2 [2a + (n – 1)d], So, when a = 1 and d = 1/2. Then, the sum of first 16 terms are;
S16 = 16/2 [2 + (16 – 1)1/2] = 8(2+15/2) = 76
And when a = 5 and d = –1/2, then the sum of first 16 terms are;
S16 = 16/2 [2(5)+ (16 – 1)(–1/2)] = 8(5/2) = 20
Also Check : Which term of the AP: 21, 18, 15, . . . is – 81? Also, is any term 0? Give reason for your answer.
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