QQuestion
The 5th term and the 9th term of an Arithmetic Progression are 4 and −12 respectively. Find:
(a) the first term
(b) the common difference
(c) the sum of 16 terms of the A.P.
✓Answer
Let the first term be a and the common difference be d.
The nth term of an A.P. is:
an = a + (n − 1)d
Given, the 5th term is 4:
⇒ a + (5 − 1)d = 4
⇒ a + 4d = 4 … (1)
Given, the 9th term is −12:
⇒ a + (9 − 1)d = −12
⇒ a + 8d = −12 … (2)
Subtracting (1) from (2):
⇒ (a + 8d) − (a + 4d) = −12 − 4
⇒ a + 8d − a − 4d = −16
⇒ 4d = −16
⇒ d = −4
Substituting d = −4 in (1):
⇒ a + 4(−4) = 4
⇒ a − 16 = 4
⇒ a = 20
(a) Therefore, the first term is 20.
(b) The common difference is −4.
(c) Sum of n terms: Sn = \frac{n}{2}[2a+(n-1)d]
⇒ S16 = \frac{16}{2}[2(20)+(16-1)(-4)]
⇒ S16 = 8[40 − 60]
⇒ S16 = 8(−20)
Therefore, S16 = −160.
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