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  4. The largest term common to the sequences 1, 11, 21, 31,...to 100 terms and 31, 36, 41, 46,..... to 100 terms is

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Q. The largest term common to the sequences $1, 11, 21, 31,...$to $100$ terms and $31, 36, 41, 46,.....$ to $100 $ terms is

Sequences and Series

Solution:

Let mth term of the first sequence be equal to the nth term of the second sequence. Then
$1 + (m - 1)10 = 31 + (n - 1)5$
$\Rightarrow 10m - 9 = 5n + 26$
$\Rightarrow 10m - 35 = 5n$
$\Rightarrow 2m -7 = n \le 100 $
$\Rightarrow 2m \le 107$
$\Rightarrow m \le 53 \frac{1}{2}$
$\therefore $ largest value of $m = 53$
largest term $= 1 + 52 \times 10 = 521$