Sequence And Series Question 289
Question: The sum of three decreasing numbers in A.P. is 27. If $ -1,,-1,,3 $ are added to them respectively, the resulting series is in G.P. The numbers are
[AMU 1999]
Options:
A) 5, 9, 13
B) 15, 9, 3
C) 13, 9, 5
D) 17, 9, 1
Show Answer
Answer:
Correct Answer: D
Solution:
Let the three terms of the series is $ a+d,a,a-d. $ \ $ a+d+a+a-d=27 $
Þ $ 3a=27 $
Þ $ a=9 $ Now, $ (a+d-1),(a-1),(a-d+3) $ are in G.P.
$ \Rightarrow $ $ {{(a-1)}^{2}}=(a+d-1)(a-d+3) $
$ \Rightarrow 64=(8+d)(12-d) $
$ \Rightarrow 64=-d^{2}+4d+96 $
$ \Rightarrow d^{2}-4d-32=0 $
$ \Rightarrow d^{2}-8d+4d-32=0 $
Þ $ (d-8)(d+4)=0 $ ,
$ \therefore d=-4,,8 $ Series is 5, 9, 13 (for d = - 4) and 17, 9, 1 (for d = 8) \ Decreasing A.P. is 17, 9, 1.
BETA
