Find the largest term common to the sequences $1,\ 11,\ 21,\ 31,\ ...$ to $100^{th}$ terms and $31,\ 36,\ 41,\ 46,\ ...$ to $100^{th}$ terms.

Given: Sequences $1,\ 11,\ 21,\ 31,\ ...$ to $100^{th}$ terms and $31,\ 36,\ 41,\ 46,\ ...$ to $100^{th}$ terms.

To do: To find the largest term common to the both sequences.

Solution:

The general term of sequence $1,\ 11,\ 21,\ 31,\ ....\ 100$ terms is $1+10n$ where $n$ goes from $0$ to $99$

The general term of sequence $31,\ 36,\ 41\ .....\ 100$ terms is $31+5k$ where $k$ goes from $0$ to $99$

For both sequences to have common term , $1+10n=31+5k$ for some values of $n,\ k$.

$\Rightarrow 10n=30+5k$

$\Rightarrow 2n=6+k$

The largest even value of $k$ is $98$, so at $k=98$, 

We get $n=52$

The largest common value is $1+10\times 52=521$
 

Updated on: 10-Oct-2022

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