Multiple choice

If ${n^{th}}$ term of the series $25 + 29 + 33 + 37 + ....$ and $3+4+6+9+13+....$ are equal, the $n$ equals

  1. $11$
  2. $12$
  3. $13$
  4. $14$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The n-th term of the first series is 25 + (n-1)4 = 4n + 21. The second series is 3, 4, 6, 9, 13, which has second differences of 1, so the n-th term is a quadratic of the form an^2 + bn + c, specifically 0.5n^2 - 0.5n + 3. Setting 4n + 21 = 0.5n^2 - 0.5n + 3 leads to n^2 - 9n - 36 = 0, which factors to (n-12)(n+3) = 0, giving n = 12.

AI explanation

The nth term of the first arithmetic progression is 25 + (n - 1)4 = 4n + 21. For the second series, the first differences are 1, 2, 3, 4, making it a quadratic sequence where the nth term Tn is found by n(n - 1)/2 + 3. Equating the two nth terms gives 4n + 21 = (n^2 - n + 6)/2. Multiplying by 2 yields 8n + 42 = n^2 - n + 6, which rearranges to n^2 - 9n - 36 = 0. Factoring this quadratic equation gives (n - 12)(n + 3) = 0, and since n must be positive, n equals 12.

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