(a)
The sum of the first three terms of an Arithmetic Progression (A.P.) is 42 and the
product of the first and third term is 52. Find the first term and the common
difference.
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Step-by-step explanation:
Let the three terms of A.P be 'a-d, a, a+d'.
The sum of the first three terms of an A.P is 42.i.e.
a-d+a+a+d=42
a−d+a+a+d=423
a=423
a=42
a=\frac{42}{3}a= 342
a=14
The first term is 14.
The product of the first and the third term is 52.i.e.
(a-d)(a+d)=52
(a−d)(a+d)=52
a^2-d^2=52
a 2 −d 2 =52
Put value of a,
(14)^2-d^2=52(14) 2 −d 2 =52196-d^2=52196−d 2 =52d^2=196-52d 2 =196−52d^2=144d 2 =144d=\sqrt{144}d= 144 d=\pm12d=±12
The common difference is 12 or -12.
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