Let the sum displaystyle ∑ n = 19(1/n (. n + 1 .) (. n + 2 .)) , written in the rational form be (p/q) (where p and q are co-prime), then the value of [(q - p/10)] is, (where [.] is the greatest integer function)

Question

Let the sum displaystyle ∑ n = 19(1/n (. n + 1 .) (. n + 2 .)) , written in the rational form be (p/q) (where p and q are co-prime), then the value of [(q - p/10)] is, (where [.] is the greatest integer function) (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

displaystyle ∑ n = 19 (1/2) (n + 2 - n/n (n + 1) (n + 2)) =(1/2) ((1/1.2) - (1/2.3)) + ((1/2.3) - (1/3.4)) + ldots + ((1/9 × 10) - (1/10 × 11)) =(1/2)((1/2) - (1/110))=(1/2)((55 - 1/110))=(27/110) ⇒ q-p=110-27=83

Q. Let the sum n=1∑9​n(n+1)(n+2)1​ , written in the rational form be qp​ (where p and q are co-prime), then the value of [10q−p​] is, (where [.] is the greatest integer function)

n=1∑9​21​n(n+1)(n+2)n+2−n​ =21​(1.21​−2.31​)+(2.31​−3.41​)+…+(9×101​−10×111​) =21​(21​−1101​)=21​(11055−1​)=11027​ ⇒q−p=110−27=83

Q. Let the sum $n = 1 \sum 9 \frac{1}{n ( n + 1 ) ( n + 2 )}$ , written in the rational form be $\frac{p}{q}$ (where $p$ and $q$ are co-prime), then the value of $[\frac{q - p}{10}]$ is, (where [.] is the greatest integer function)