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in Geometric Progressions by (23.5k points)
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If a, b, c, d are in G.P, then (a + b + c + d)2 is equal to

(a) (a + b)2 + (c + d)2 + 2(b + c)2 

(b) (a + b)2 + (c + d)2 + 2(a + c)2 

(c) (a + b)2 + (c + d)2 + 2(b + d)2 

(d) (a + b)2 + (c + d)2 + (b + c)2

1 Answer

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Best answer

(a) (a + b)2 + (c + d)2 + 2(b + c)2 

a, b, c, d are in G.P. 

⇒ b = ar, c = ar2, d = ar3, where r is the common ratio of the G.P. 

LHS = (a + b + c + d)2 = (a + ar + ar2 + ar3)2 

= {a (1 + r + r2 + r3)}2 = {a ((1 + r) + r2 (1 + r))}2 

= {a (1 + r) (1 + r2)}2 = a2 (1 + r)2 (1 + r2)2 

= a2 (1 + r)2 (r4 + 2r2 + 1) 

= a2 (r4 (1 + r)2 + 2r2(1 + r)2 + (1 + r)2

= a2r4 (1 + r)2 + 2a2r2(1 + r)2 + a2(1 + r)2 

= (ar2 + ar3)2 + 2(ar + ar2)2 + (a + ar)2 

= (c + d)2 + 2(b + c)2 + (a + b)2.

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