Multiple choice

The sum of the first $25$ terms of an arithmetic sequence is $1,400$, and the $25th$ term is $104$. F ind the value of $a_{2} - a_{1}$ where the first term of the sequence is $a_{1}$ and the second term is $a_{2}$.

  1. $-3$
  2. $2$
  3. $4$
  4. $5$
  5. $8$
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C Correct answer
Explanation

The sum of an arithmetic sequence is given by S_n = (n/2)(a_1 + a_n). Substituting the given values, 1400 = (25/2)(a_1 + 104), which simplifies to a_1 = 8. Using the formula for the 25th term, 104 = 8 + 24*d, we find the common difference d = a_2 - a_1 = 4.

AI explanation

The sum of the first n terms of an arithmetic sequence is S_n = n/2 * (a_1 + a_n). Given S_25 = 1400 and a_25 = 104, we have 1400 = 25/2 * (a_1 + 104), which solves to a_1 = 8. Using the nth term formula a_n = a_1 + (n-1)d, we get 104 = 8 + 24d, giving d = 4. The value of a_2 - a_1 is simply the common difference d, which is 4.

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