The coefficient of x 48 in ( 1 + x ) + 2 ( 1 + x ) 2 + 3 ( 1 + x ) 3 + ? + 100 ( 1 + x ) 100 is equal to [2026]

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Q.
The coefficient of $x^{48}$ in $(1 + x) + 2 {(1 + x)}^{2} + 3 {(1 + x)}^{3} + \dots + 100 {(1 + x)}^{100}$ is equal to [2026]

1 $100 \cdot {}^{100}{C_{49}} - {}^{100}{C_{48}}$

2 ${}^{100}{C_{50}} + {}^{101}{C_{49}}$

3 $100 \cdot {}^{101}{C_{49}} - {}^{101}{C_{50}}$

4 $100 \cdot {}^{100}{C_{49}} - {}^{100}{C_{50}}$

Ans.
(3)

$Let 1 + x = r$

$∴$ $S = 1 \cdot r + 2 \cdot r^{2} + 3 \cdot r^{3} + \dots + 100 r^{100} . . . (1)$
$(AGP)$

$r S = 1 \cdot r^{2} + 2 \cdot r^{3} + \dots + 99 r^{100} + 100 r^{101} . . . (2)$

$(1) - (2) gives$

$S = - \frac{{(1 + x)}^{101}}{x^{2}} + \frac{1}{x^{2}} + \frac{100 {(1 + x)}^{101}}{x}$

$∴$ $coefficient of x^{48} in S$

$= - coefficient of x^{48} in \frac{{(1 + x)}^{101}}{x^{2}} + 100 . Coefficient of x^{48} in \frac{{(1 + x)}^{101}}{x}$

$= 100 . {}^{101}{C_{49}} - {}^{101}{C_{50}}$

Solution

Q.
The coefficient of $x^{48}$ in $(1 + x) + 2 {(1 + x)}^{2} + 3 {(1 + x)}^{3} + \dots + 100 {(1 + x)}^{100}$ is equal to [2026]

1 $100 \cdot {}^{100}{C_{49}} - {}^{100}{C_{48}}$

2 ${}^{100}{C_{50}} + {}^{101}{C_{49}}$

3 $100 \cdot {}^{101}{C_{49}} - {}^{101}{C_{50}}$

4 $100 \cdot {}^{100}{C_{49}} - {}^{100}{C_{50}}$

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Solution

Q. The coefficient of x48 in (1+x)+2(1+x)2+3(1+x)3+⋯+100(1+x)100 is equal to [2026]

1 100·C49100-C48100

2 C50100+C49101

3 100·C49101-C50101

4 100·C49100-C50100

Ans. (3) Let 1+x=r ∴S=1·r+2·r2+3·r3+⋯+100r100 ...(1) (AGP) rS=1·r2+2·r3+⋯+99r100+100r101 ...(2) (1)-(2) gives S=-(1+x)101x2+1x2+100(1+x)101x ∴coefficient of x48 in S =-coefficient of x48 in (1+x)101x2+100.Coefficient of x48 in (1+x)101x =100.C49101-C50101

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Q.
The coefficient of $x^{48}$ in $(1 + x) + 2 {(1 + x)}^{2} + 3 {(1 + x)}^{3} + \dots + 100 {(1 + x)}^{100}$ is equal to [2026]

1 $100 \cdot {}^{100}{C_{49}} - {}^{100}{C_{48}}$

2 ${}^{100}{C_{50}} + {}^{101}{C_{49}}$

3 $100 \cdot {}^{101}{C_{49}} - {}^{101}{C_{50}}$

4 $100 \cdot {}^{100}{C_{49}} - {}^{100}{C_{50}}$