Question

4 which explicit formula describes the sequence -10, -3, 4, 11, 18, ...? a $a_n = 7n + 17$ b $a_n = 7n - 17$ c $a_n = 10n - 7$ d $a_n = -10n - 7$

Explanation:

Step1: Identify sequence type

The sequence \(-10, -3, 4, 11, 18, \dots\) has a common difference. Calculate the difference between consecutive terms: \(-3 - (-10) = 7\), \(4 - (-3) = 7\), \(11 - 4 = 7\), \(18 - 11 = 7\). So it's an arithmetic sequence with first term \(a_1 = -10\) and common difference \(d = 7\).

Step2: Recall arithmetic sequence formula

The explicit formula for an arithmetic sequence is \(a_n = a_1 + (n - 1)d\). Substitute \(a_1 = -10\) and \(d = 7\):

$$
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$$

Step3: Verify with options

Check each option:

• Option A: \(a_n = 7n + 17\). For \(n = 1\), \(7(1) + 17 = 24

eq -10\). Eliminate.

• Option B: \(a_n = 7n - 17\). For \(n = 1\), \(7(1) - 17 = -10\) (matches \(a_1\)). For \(n = 2\), \(7(2) - 17 = -3\) (matches \(a_2\)). For \(n = 3\), \(7(3) - 17 = 4\) (matches \(a_3\)). This works.
• Option C: \(a_n = 10n - 7\). For \(n = 1\), \(10(1) - 7 = 3

eq -10\). Eliminate.

• Option D: \(a_n = -10n - 7\). For \(n = 1\), \(-10(1) - 7 = -17

eq -10\). Eliminate.

Answer:

B. \(a_n = 7n - 17\)