In an arithmetic sequence, the first term is 2 and the second term is 5.(a) Find the common difference. [2](b) Find the eighth term. [2](c) Find the sum of the first eight terms of the sequence.
Solution
(a) The common difference in an arithmetic sequence is found by subtracting the first term from the second term. So, 5 - 2 = 3. The common difference is 3.
(b) The nth term in an arithmetic sequence can be found using the formula: a + (n - 1)d, where a is the first term, n is the term number, and d is the common difference. Substituting the given values, we get: 2 + (8 - 1)3 = 2 + 21 = 23. So, the eighth term is 23.
(c) The sum of the first n terms of an arithmetic sequence can be found using the formula: n/2 * (2a + (n - 1)d), where a is the first term, n is the number of terms, and d is the common difference. Substituting the given values, we get: 8/2 * (2*2 + (8 - 1)3) = 4 * (4 + 21) = 4 * 25 = 100. So, the sum of the first eight terms is 100.
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