QQuestion
The nth term of an Arithmetic Progression (A.P.) is given by the relation Tn = 6(7 − n). Find:
(a) its first term and common difference.
(b) the sum of its first 25 terms.
✓Answer
Given, Tn = 6(7 − n).
(a) Put n = 1:
T₁ = 6(7 − 1)
⇒ T₁ = 6 × 6
⇒ T₁ = 36
Therefore, the first term a = 36.
Put n = 2:
T₂ = 6(7 − 2)
⇒ T₂ = 6 × 5
⇒ T₂ = 30
Common difference, d = T₂ − T₁
⇒ d = 30 − 36
⇒ d = −6
(b) The sum of n terms of an A.P. is:
Sₙ = \frac{n}{2}[2a+(n-1)d]
For n = 25:
S₂₅ = \frac{25}{2}[2(36)+(25-1)(-6)]
⇒ S₂₅ = \frac{25}{2}[72+24(-6)]
⇒ S₂₅ = \frac{25}{2}[72-144]
⇒ S₂₅ = \frac{25}{2}(-72)
⇒ S₂₅ = 25 × (−36)
⇒ S₂₅ = −900
Therefore, the first term is 36, the common difference is −6, and the sum of the first 25 terms is −900.
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