If T= displaystyle∑r=118 ((-1)r-1 ⋅ r ⋅ 18 Cr/(r+1)) then ((1/T)-10) is equal to

Question

If T= displaystyle∑r=118 ((-1)r-1 ⋅ r ⋅ 18 Cr/(r+1)) then ((1/T)-10) is equal to (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

(r/(r+1)) ⋅ 18 Cr=(1-(1/r+1)) ⋅ 18 Cr = 18 Cr-(⌊ 18/(r+1)⌊ r⌊ 18-r) = 18 Cr-(1/19) 19 Cr+1 ∴ T= displaystyle∑r=118(-1)r-1 ⋅ (r/(r+1)) ⋅ 18 Cr = 18 C1- 18 C2+ 18 C3- ldots = 18 C0-(1/19)(- 19 C0+ 19 C1) =1+(1/19)-1=(1/19) ⇒ (1/T)=19

Q. If T=r=1∑18​(r+1)(−1)r−1⋅r⋅18Cr​​ then (T1​−10) is equal to

(r+1)r​⋅18Cr​=(1−r+11​)⋅18Cr​ =18Cr​−(r+1)⌊r⌊18−r⌊18​ =18Cr​−191​19Cr+1​ ∴T=r=1∑18​(−1)r−1⋅(r+1)r​⋅18Cr​ =18C1​−18C2​+18C3​−… =18C0​−191​(−19C0​+19C1​) =1+191​−1=191​ ⇒T1​=19

Q. If $T = r = 1 \sum 18 \frac{( - 1 ) ^{r - 1} \cdot r \cdot ^{18} C _{r}}{( r + 1 )}$ then $(\frac{1}{T} - 10)$ is equal to