Q. If $a,\ b,\ c,\ d$ are any four consecutive coefficients of any expanded binomial, then $\frac{a + b}{a},\frac{b + c}{b},\frac{c + d}{c}$ are in
A) AP
B) GP
C) HP
D) None of these
Explanation
(c)
Let $a =^{n}C_{r - 1},\ b =^{n}C_{r},\$and $d =^{n}C_{r + 2}$
Substituting these values in given expression
$\frac{a + b}{a},\frac{b + c}{b},\frac{c + d}{c}$
$\Rightarrow \ \frac{\ ^{n}C_{r - 1} +^{n}C_{r}}{\ ^{n}C_{r - 1}},\frac{\ ^{n}C_{r} +^{n}C_{r + 1}}{\ ^{n}C_{r}},\frac{\ ^{n}C_{r + 1} +^{n}C_{r + 2}}{\ ^{n}C_{r + 1}}$
$\Rightarrow \ \ \frac{\ ^{n + 1}C_{r}}{\ ^{n}C_{r - 1}},\ \frac{\ ^{n + 1}C_{r + 1}}{\ ^{n}C_{r}},\frac{\ ^{n + 1}C_{r + 2}}{\ ^{n}C_{r + 1}},$
$\Rightarrow \ \$It is in HP.
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.