Arithmetic

What is the sum of the following sequence?

111!+12!+112!+13!+113!+14!+⋯+131!+32!\displaystyle {\frac{1}{11!+12!} + \frac{1}{12!+13!} +\frac{1}{13!+14!} +\cdots + \frac{1}{31!+32!}}

a) 111!−132!\displaystyle {\frac{1}{11!} - \frac{1}{32!} }

b) 112!−132!\displaystyle {\frac{1}{12!} - \frac{1}{32!} }

c) 110!−132!\displaystyle {\frac{1}{10!} - \frac{1}{32!} }

d) 112!−133!\displaystyle {\frac{1}{12!} - \frac{1}{33!} }

Ans: d) 112!−133!\displaystyle {\frac{1}{12!} - \frac{1}{33!} }

Explanation:

Concept of n!n! is required to solve this type of problem.

n!=n×(n−1)!=n×(n−1)×(n−2)! n! =n\times (n-1)!=n\times (n-1)\times (n-2)! etc.

111!+12!=111!+12×11!=1(1+12)×11!=113×11!=1213×12×11!=1213!=13−113!=1313!−113!=1313×12!−113!=112!−113! \begin{align*} \frac{1}{11!+12!}&=\frac{1}{11!+12\times 11!}\\ &=\frac{1}{(1+12)\times 11!}\\ &=\frac{1}{13\times 11!}\\ &=\frac{12}{13\times 12\times 11!}\\ &=\frac{12}{13!}\\ &=\frac{13-1}{13!}\\ &=\frac{13}{13!}-\frac{1}{13!}\\ &=\frac{13}{13\times 12!}-\frac{1}{13!}\\ &=\frac{1}{12!}-\frac{1}{13!}\\ \end{align*}
112!+13!=112!+13×12!=1(1+13)×12!=114×12!=1314×13×12!=1314!=14−114!=113!−114! \begin{align*} \frac{1}{12!+13!}&=\frac{1}{12!+13\times 12!}\\ &=\frac{1}{(1+13)\times 12!}\\ &=\frac{1}{14\times 12!}\\ &=\frac{13}{14\times 13\times 12!}\\ &=\frac{13}{14!}\\ &=\frac{14-1}{14!}\\ &=\frac{1}{13!}-\frac{1}{14!}\\ \end{align*}

Similarly, 113!+14!=114!−115! \begin{align*} \frac{1}{13!+14!} &=\frac{1}{14!}-\frac{1}{15!}\\ \end{align*}

And, 131!+32!=132!−133! \begin{align*} \frac{1}{31!+32!} &=\frac{1}{32!}-\frac{1}{33!}\\ \end{align*}

Now writing the full expression:

111!+12!+112!+13!+113!+14!+⋯+131!+32!=112!−113!+113!−114!+114!−115!+⋯+132!−133!=112!−133! \begin{align*} &\frac{1}{11!+12!} + \frac{1}{12!+13!} +\frac{1}{13!+14!} +\cdots + \frac{1}{31!+32!}\\ &=\frac{1}{12!}-\frac{1}{13!} + \frac{1}{13!}-\frac{1}{14!} + \frac{1}{14!}-\frac{1}{15!} + \cdots + \frac{1}{32!}-\frac{1}{33!}\\ &=\frac{1}{12!}-\frac{1}{33!} \end{align*}

Except first and last terms, all will be cancelled out.

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4 thoughts on “1/(11!+12!)+1/(12!+13!)+1/(13!+14!)+⋯+1/(31!+32!)”

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