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

Entered answer:

✅ Correct Answer: 967

Let's start by understanding what these sequences actually look like.

For sequence $a_n = 13 + 6(n-1)$ :

$a_1 = 13 + 6(1-1) = 13 + 0 = 13$

$a_2 = 13 + 6(2-1) = 13 + 6 = 19$

$a_3 = 13 + 6(3-1) = 13 + 12 = 25$

$a_4 = 13 + 6(4-1) = 13 + 18 = 31$

So sequence $a_n$ : $13, 19, 25, 31, 37, 43, 49, 55, 61, 67, 73, 79, 85, ...$

For sequence $b_n = 15 + 7(n-1)$ :

$b_1 = 15 + 7(1-1) = 15 + 0 = 15$

$b_2 = 15 + 7(2-1) = 15 + 7 = 22$

$b_3 = 15 + 7(3-1) = 15 + 14 = 29$

$b_4 = 15 + 7(4-1) = 15 + 21 = 36$

So sequence $b_n$ : $15, 22, 29, 36, 43, 50, 57, 64, 71, 78, 85, ...$

We can spot the common terms: 43, 85, ...

Key Insight: Both sequences are arithmetic progressions (AP). When we want common terms between two APs, they form another AP!

The first common term is 43.

Why do we use LCM?

Sequence $a_n$ increases by 6 each time

Sequence $b_n$ increases by 7 each time

The gap between consecutive common terms will be LCM(6,7) = 42

Think of it this way: Starting from 43, the next common term will be when both sequences "meet again." This happens after LCM(6,7) = 42 units.

Therefore, all common terms follow the pattern: $43, 43+42, 43+2×42, 43+3×42, ...$

General form: Common terms = $42k + 43$ where $k = 0, 1, 2, 3, ...$

We need the largest value of $42k + 43$ that is still less than 1000.

$42k + 43 < 1000$

$42k < 957$

$k < \tfrac{957}{42}$

$k < 22.78...$

Since $k$ must be a whole number, the largest possible value is $k = 22$ .

Final Answer: $42(22) + 43 = 924 + 43 = 967$

967

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Let ana_nan​ and bnb_nbn​ be two sequences such that an=13+6(n−1)a_n = 13 + 6 (n - 1)an​=13+6(n−1) and bn=15+7(n−1)b_n = 15 + 7 (n - 1)bn​=15+7(n−1) for all natural numbers nnn. Then, the largest three digit integer that is common to both these sequences, is

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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