Question

pls tell me procedure What is the value of ((1-1/6) (1-1/7)........(1-(1/n+4)) (1-(1/n+5)))?..(ans: 5/(n+5))

Answer 1

\[\prod_{k =2}^{n}\left(1-\frac{1}{k+4}\right)\left(1-\frac{1}{k+5}\right)\]\[1-\frac{1}{k+4}=\frac{k+4}{k+4}-\frac{1}{k+4}=\frac{k+3}{k+4}\]
\[1-\frac{1}{k+5}=\frac{k+5}{k+5}-\frac{1}{k+5}=\frac{k+4}{k+5}\]So we really want:\[\prod_{k=2}^n \frac{k+3}{k+4}\cdot \frac{k+4}{k+5}=\prod_{k=2}^n\frac{k+3}{n+5}=\frac{5}{7}\cdot\frac{6}{8}\cdot\frac{7}{9}\cdots\frac{n+3}{n+5}\]\[=\frac{5\cdot6}{(n+4)(n+5)}\]

Answer 2

are you sure the answer is 5/(n+5)? its not cancelling out right...

Answer 3

yes sirr!

Answer 4

im am getting the question right? its:\[\left(1-\frac{1}{6}\right)\left(1-\frac{1}{7}\right)\left(1-\frac{1}{7}\right)\left(1-\frac{1}{8}\right)\ldots\left(1-\frac{1}{n+4}\right)\left(1-\frac{1}{n+5}\right)\]

Answer 5

no no no, im not, its this:\[\left(1-\frac{1}{6}\right)\left(1-\frac{1}{7}\right)\left(1-\frac{1}{8}\right)\ldots \left(1-\frac{1}{n+4}\right)\left(1-\frac{1}{n+5}\right)\]

Answer 6

which is equal to:\[\frac{5}{6}\cdot\frac{6}{7}\cdot\frac{7}{8}\cdots \frac{n+3}{n+4}\cdot \frac{n+4}{n+5}\]can you see whats going to happen from there?

Answer 7

\[[{(1-1/6)(1-1/7)....(1-1/(n+4)}(1-1/(n+5)]\]

Answer 8

[{(1−1/6)(1−1/7)....(1−1/(n+4)}(1−1/(n+5)] i think the question was in ths mannr...i m getting ur point sir!!oka i m tring again