Write ∑ n+rCr r∈(0,m) in the simplified form. ​

Question

\(\displaystyle\sum_{r=1}^{m}\)^n+rC_r 

Write ∑^n+rC_r r∈(0,m) in the simplified form.

Answer 1

Given,

∑^n+rC_r r∈(0,m)

We know : 

ⁿC_r + ⁿC_r-1 = ⁿ⁺¹C_{r ....}(1)

\(\sum_{r=0}^{m} \) ^n+rC_r = ⁿC₀ + ⁿ⁺¹C₁ + ⁿ⁺²C₂ + ⁿ⁺³C₃ + . . . . . . + ^n+mC_n+1

\(\sum_{r=0}^{m} \)^n+rC_r = ⁿ⁺¹C₀ + ⁿ⁺¹C₁ + ⁿ⁺²C₂ + ⁿ⁺³C₃ + . . . . . . + ^n+mC_n+1 

⇒ (ⁿC₀ = ⁿ⁺¹C₀)

Using equation (1),

 \(\sum_{r=0}^{m} \)^n+rC_r = ⁿ⁺²C₁ + ⁿ⁺²C₂ + ⁿ⁺³C₃ + . . . . . . + ^n+mC_n+1

 \(\sum_{r=0}^{m} \)^n+rC_r = ⁿ⁺³C₂ + ⁿ⁺³C₃ + . . . . . . + ^n+mC_n+1

Proceeding in the same way :

 \(\sum_{r=0}^{m} \)^n+rC_r = ^n+mC_m-1 + ^n+mC_m = ^n+m+1C_n+1

 \(\sum_{r=0}^{m} \)^n+rC_r = ^n+m+1C_n+1