Let $$a_{n}$$ and $$b_{n}$$ be two sequences such that $$a_{n}=13+6(n-1)$$ and $$b_{n}=15+7(n-1)$$ for all natural numbers n. Then, the largest three digit integer that is common to both these sequences, is
Correct Answer: 967
It is given thatĀ $$a_{n}=13+6(n-1)$$, which can be written asĀ $$a_n=13+6n-6\ =\ 7+6n$$
Similarly,Ā $$b_{n}=15+7(n-1)$$, which can be written asĀ $$b_n=15+7n-7\ =\ 8+7n$$
The common differences are 6, and 7, respectively, The common difference of terms that exists in both series is l.c.m (6, 7) = 42
The first common term of the first two series is 43 (by inspection)
Hence, we need to find the mth term, which is less than 1000, and the largest three-digit integer, and exists in both series.
$$t_m=a+\left(m-1\right)d\ <\ 1000$$
=> $$43+\left(m-1\right)42\ <\ 1000$$
=> $$\left(m-1\right)42\ <\ 957$$
=> m-1 < 22.8 => m < 23.8 => m = 23
Hence, the 23rd term isĀ $$43+22\times\ 42\ =\ 967$$
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