Sequence And Series Question 303
Question: If the sum of three numbers of a arithmetic sequence is 15 and the sum of their squares is 83, then the numbers are
[MP PET 1985]
Options:
A) 4, 5, 6
B) 3, 5, 7
C) 1, 5, 9
D) 2, 5, 8
Show Answer
Answer:
Correct Answer: B
Solution:
Let three numbers are $ a-d,\ a,\ a+d $ . We get $ a-d+a+a+d=15 $
$ \Rightarrow $ $ a=5 $ and $ {{(a-d)}^{2}}+a^{2}+{{(a+d)}^{2}}=83 $
$ \Rightarrow $ $ a^{2}+d^{2}-2ad+a^{2}+a^{2}+d^{2}+2ad=83 $
$ \Rightarrow $ $ 2(a^{2}+d^{2})+a^{2}=83 $ Putting $ a=5 $
$ \Rightarrow $ $ 2(25+d^{2})+25=83 $
$ \Rightarrow $ $ 2d^{2}=8 $
$ \Rightarrow $ $ d=2 $ Thus numbers are 3, 5, 7. Trick: Since $ 3+5+7=15 $ and $ 3^{2}+5^{2}+7^{2}=83 $ .
BETA
