A series is given with one term missing. Select the correct alternative from the given ones that will complete the series. -6, -15, -57, -168, -669, -2004, ? , -24036
-8013
This is a multiplicative number series in which each term is built from the previous one by a multiplier and a fixed addition.
The terms are \(-6,\ -15,\ -57,\ -168,\ -669,\ -2004,\ ?,\ -24036\).
Test a rule of the form “multiply then add 3”, alternating the multiplier between 3 and 4.
\(-6 \times 3 + 3 = -15\) and \(-15 \times 4 + 3 = -57\).
\(-57 \times 3 + 3 = -168\), then \(-168 \times 4 + 3 = -669\) and \(-669 \times 3 + 3 = -2004\), so the pattern holds throughout.
The next operation in the alternation is \(\times 4 + 3\), applied to \(-2004\): \(-2004 \times 4 + 3 = -8016 + 3 = -8013\).
Check the following term with the \(\times 3 + 3\) step: \(-8013 \times 3 + 3 = -24039 + 3 = -24036\), which is exactly the last given term.
The concept is an alternating geometric-type recurrence with a constant offset. Hence the missing term is -8013.
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