Sequences And Series Question 1
Question 1 - 2024 (01 Feb Shift 1)
Let $3, \mathrm{a}, \mathrm{b}, \mathrm{c}$ be in A.P. and $3, \mathrm{a}-1, \mathrm{~b}+1, \mathrm{c}+9$ be in G.P. Then, the arithmetic mean of $a, b$ and $c$ is :
(1) -4
(2) -1
(3) 13
(4) 11
Show Answer
Answer (4)
Solution
$3, \mathrm{a}, \mathrm{b}, \mathrm{c} \rightarrow$ A.P $\quad \Rightarrow 3,3+\mathrm{d}, 3+2 \mathrm{~d}, 3+3 \mathrm{~d}$
$3, \mathrm{a}-1, \mathrm{~b}+1, \mathrm{c}+9 \rightarrow$ G.P $\Rightarrow 3,2+\mathrm{d}, 4+2 \mathrm{~d}, 12+3 \mathrm{~d}$
$\mathrm{a}=3+\mathrm{d} \quad(2+d)^{2}=3(4+2 d)$
$\mathrm{b}=3+2 \mathrm{~d} \quad \mathrm{~d}=4,-2$
$\mathrm{c}=3+3 \mathrm{~d}$
If $\mathrm{d}=4 \quad$ G.P $\Rightarrow 3,6,12,24$
$\mathrm{a}=7$
$\mathrm{b}=11$
$\mathrm{c}=15$
$\frac{a+b+c}{3}=11$
