3, −2, −12, −32, −72, ?
Series Differences
The given number series is: 3, −2, −12, −32, −72, ?
Calculate the differences between consecutive terms to identify a pattern.
| Term Pair | Difference ($a_{n+1} - a_n$) |
| 3 and −2 | $-2 - 3 = -5$ |
| −2 and −12 | $-12 - (-2) = -10$ |
| −12 and −32 | $-32 - (-12) = -20$ |
| −32 and −72 | $-72 - (-32) = -40$ |
Difference Pattern
The calculated differences are: $-5, -10, -20, -40$.
Notice that each difference is twice the previous difference:
- $-10 = -5 \times 2$
- $-20 = -10 \times 2$
- $-40 = -20 \times 2$
Following this pattern, the next difference is $-40 \times 2 = -80$.
Next Term Calculation
To find the missing term (the next term in the series), add this next difference ($-80$) to the last term of the series ($-72$).
Missing Term = Last Term + Next Difference
Missing Term = $-72 + (-80)$
Missing Term = $-72 - 80$
Missing Term = $-152$
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