Sequences & Series Question - None

Question

Find the number of common terms to the two sequences 17, 21, 25, ..., 417 and 16, 21, 26, ..., 466.

Accepted Answer

Step 1: Set up the equation for common terms.

Let the common term be: 17 + 4(m-1) = 16 + 5(n-1)

Simplifying: 4m - 4 + 17 = 5n - 5 + 16

4m + 13 = 5n + 11

4m - 5n = -2

Step 2: Solve the linear Diophantine equation 4m - 5n = -2.

One solution: m = 2, n = 2 (check: 4(2) - 5(2) = 8 - 10 = -2 ✓)

General solution: m = 2 + 5t, n = 2 + 4t, where t is any integer.

Step 3: Find the range of t using sequence constraints.

For first sequence: 17 + 4(m-1) ≤ 417

4m ≤ 404 → m ≤ 101

So: 2 + 5t ≤ 101 → t ≤ 19.8 → t ≤ 19

For second sequence: 16 + 5(n-1) ≤ 466

5n ≤ 455 → n ≤ 91

So: 2 + 4t ≤ 91 → t ≤ 22.25 → t ≤ 22

Also, m ≥ 1 and n ≥ 1:

2 + 5t ≥ 1 → t ≥ -0.2 → t ≥ 0

2 + 4t ≥ 1 → t ≥ -0.25 → t ≥ 0

Step 4: Count valid values of t.

Combined constraint: 0 ≤ t ≤ 19

Number of values: t ∈ {0, 1, 2, ..., 19}

∴ Answer: 20

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