Consider a sequence of 1001 terms as ( 1001 C0/1 ⋅ 2 ⋅ 3 ⋅ 4), ( 1001 C1/2 ⋅ 3 ⋅ 4 ⋅ 5), ( 1001 C2/3 ⋅ 4 ⋅ 5 ⋅ 6), ldots ( 1001 C1000/1001 ⋅ 1002 ⋅ 1003 ⋅ 1004)

Question

Consider a sequence of 1001 terms as ( 1001 C0/1 ⋅ 2 ⋅ 3 ⋅ 4), ( 1001 C1/2 ⋅ 3 ⋅ 4 ⋅ 5), ( 1001 C2/3 ⋅ 4 ⋅ 5 ⋅ 6), ldots ( 1001 C1000/1001 ⋅ 1002 ⋅ 1003 ⋅ 1004) (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Tr+1=( 1001 Cr/(r+1)(r+2)(r+3)(r+4)) =( 1004 Cr+4/1001 ⋅ 1002 ⋅ 1003 ⋅ 1004) is greatest when r=498 ∴ 499 th term is greatest.

Q. Consider a sequence of $1001$ terms as
$\frac{^{1001} C _{0}}{1 \cdot 2 \cdot 3 \cdot 4}, \frac{^{1001} C _{1}}{2 \cdot 3 \cdot 4 \cdot 5}, \frac{^{1001} C _{2}}{3 \cdot 4 \cdot 5 \cdot 6}, \dots \frac{^{1001} C _{1000}}{1001 \cdot 1002 \cdot 1003 \cdot 1004}$

$T_{r + 1} = \frac{^{1001} C _{r}}{( r + 1 ) ( r + 2 ) ( r + 3 ) ( r + 4 )}$
$= \frac{^{1004} C _{r + 4}}{1001 \cdot 1002 \cdot 1003 \cdot 1004}$ is greatest when $r = 498$
$∴ 499$ th term is greatest.

Q. Consider a sequence of 1001 terms as 1⋅2⋅3⋅41001C0​​,2⋅3⋅4⋅51001C1​​,3⋅4⋅5⋅61001C2​​,…1001⋅1002⋅1003⋅10041001C1000​​

Tr+1​=(r+1)(r+2)(r+3)(r+4)1001Cr​​ =1001⋅1002⋅1003⋅10041004Cr+4​​ is greatest when r=498 ∴499 th term is greatest.

Q. Consider a sequence of $1001$ terms as
$\frac{^{1001} C _{0}}{1 \cdot 2 \cdot 3 \cdot 4}, \frac{^{1001} C _{1}}{2 \cdot 3 \cdot 4 \cdot 5}, \frac{^{1001} C _{2}}{3 \cdot 4 \cdot 5 \cdot 6}, \dots \frac{^{1001} C _{1000}}{1001 \cdot 1002 \cdot 1003 \cdot 1004}$

$T_{r + 1} = \frac{^{1001} C _{r}}{( r + 1 ) ( r + 2 ) ( r + 3 ) ( r + 4 )}$
$= \frac{^{1004} C _{r + 4}}{1001 \cdot 1002 \cdot 1003 \cdot 1004}$ is greatest when $r = 498$
$∴ 499$ th term is greatest.