Write - r -0 m Crn - r in the simplified form-

We know: Crn+Cr 1n=Crn+1 #8721;r=0mCrn+r=C0n+C1n+1+C2n+2+C3n+3+...+Cmn+m #8757; #160;C0n=C0n+1 #8756; #160; #8721;r=0mCrn+r=C0n+1+C1n+1+C2n+2+C3n+ ...

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Byju's Answer

Standard XI

Economics

Total Costs in the Short Run

Write ∑ r =0 ...

Question

Write $\sum_{r = 0}^{m} {}^{n + r}C_{r}$ in the simplified form.

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Solution

We know:
${}^{n}C_{r} + {}^{n}C_{r - 1} = {}^{n + 1}C_{r}$

$\sum_{r = 0}^{m} {}^{n + r}C_{r} = {}^{n}C_{0} + {}^{n + 1}C_{1} + {}^{n + 2}C_{2} + {}^{n + 3}C_{3} + . . . + {}^{n + m}C_{m} ∵ {}^{n}C_{0} = {}^{n + 1}C_{0} ∴ \sum_{r = 0}^{m} {}^{n + r}C_{r} = {}^{n + 1}C_{0} + {}^{n + 1}C_{1} + {}^{n + 2}C_{2} + {}^{n + 3}C_{3} + . . . + {}^{n + m}C_{m} Using {}^{n}C_{r - 1} + {}^{n}C_{r} = {}^{n + 1}C_{r} : \Rightarrow \sum_{r = 0}^{m} {}^{n + r}C_{r} = {}^{n + 2}C_{1} + {}^{n + 2}C_{2} + {}^{n + 3}C_{3} + . . . + {}^{n + m}C_{m} \Rightarrow \sum_{r = 0}^{m} {}^{n + r}C_{r} = {}^{n + 3}C_{2} + {}^{n + 3}C_{3} + . . . + {}^{n + m}C_{m}$

Proceeding in the same way:

$\sum_{r = 0}^{m} {}^{n + r}C_{r} = {}^{n + m}C_{m - 1} + {}^{n + m}C_{m} = {}^{n + m + 1}C_{m} \Rightarrow \sum_{r = 0}^{m} {}^{n + r}C_{r} = {}^{n + m + 1}C_{m}$

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Write ∑r=0m Crn+r in the simplified form.

Write $\sum_{r = 0}^{m} {}^{n + r}C_{r}$ in the simplified form.