Multiple choice

Given an arithmetic progression with its first three terms as $2t, 5t-1$ and $6t+2$, for some real number $t$. Find the value of the fourth term of the sequence.

  1. $4$
  2. $8$
  3. $10$
  4. $16$
  5. $19$
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E Correct answer
Explanation

In an AP, the difference between consecutive terms is constant: (5t-1) - 2t = (6t+2) - (5t-1). So, 3t - 1 = t + 3, which means 2t = 4, so t = 2. The terms are 2(2)=4, 5(2)-1=9, 6(2)+2=14. The common difference is 5. The fourth term is 14 + 5 = 19.

AI explanation

Because the sequence is an arithmetic progression, the difference between the second and first terms equals the difference between the third and second terms, so (5t - 1) - 2t = (6t + 2) - (5t - 1). Simplifying this equation gives 3t - 1 = t + 3, which solves to t = 2. The first three terms become 4, 9, and 14, making the common difference 5. Therefore, the fourth term is 14 + 5 = 19.

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