If the coefficients of r th , ( r + 1 ) th and ( r + 2 ) th terms in the binomial expansion of ( 1 + y ) m are in A.P., then m and r satisfy the equation

Question

If the coefficients of $r^{th}, {(r + 1)}^{th}$ and ${(r + 2)}^{th}$ terms in the binomial expansion of ${(1 + y)}^{m}$ are in A.P., then $m$ and $r$ satisfy the equation

Smart Achievers · Accepted Answer

(2)

$T_{r + 1} = {}^{m}C_{r} y^{r}$

$∴$ ${}^{m}C_{r - 1} + {}^{m}C_{r + 1} = 2 \times {}^{m}C_{r}$

$\Rightarrow$ $\frac{m!}{(r - 1)! (m - r + 1)!} + \frac{m!}{(r + 1)! (m - r - 1)!} = 2 \frac{m!}{r! (m - r)!}$

$\Rightarrow$ $\frac{r (r + 1)}{(r + 1)! (m - r + 1)!} + \frac{(m - r + 1) (m - r)}{(r + 1)! (m - r + 1)!} = \frac{2 (r + 1) (m - r + 1)}{(r + 1)! (m - r + 1)!}$

$\Rightarrow$ $r (r + 1) + (m - r + 1) (m - r) = 2 (r + 1) (m - r + 1)$

$\Rightarrow$ $r (r + 1) + {(m - r)}^{2} + m - r - 2 (r + 1) (m - (r - 1)) = 0$

$\Rightarrow$ $r (r + 1) + m^{2} + r^{2} - 2 m r + m - r - 2 (r^{2} - 1) - 2 m (r + 1) = 0$

$\Rightarrow$ $m^{2} - m (4 r + 1) + 4 r^{2} - 2 = 0$

Solution

Q.
If the coefficients of $r^{th}, {(r + 1)}^{th}$ and ${(r + 2)}^{th}$ terms in the binomial expansion of ${(1 + y)}^{m}$ are in A.P., then $m$ and $r$ satisfy the equation

1 $m^{2} - m (4 r - 1) + 4 r^{2} + 2 = 0$

2 $m^{2} - m (4 r + 1) + 4 r^{2} - 2 = 0$

3 $m^{2} - m (4 r + 1) + 4 r^{2} + 2 = 0$

4 $m^{2} - m (4 r - 1) + 4 r^{2} - 2 = 0$

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Solution

Q. If the coefficients of rth, (r+1)th and (r+2)th terms in the binomial expansion of (1+y)m are in A.P., then m and r satisfy the equation

1 m2-m(4r-1)+4r2+2=0

2 m2-m(4r+1)+4r2-2=0

3 m2-m(4r+1)+4r2+2=0