Series (Number & Letter)
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How every series is cracked
Gap between terms
d_n = a_{n+1} - a_n
Write these gaps under the series. If they are not clear, take gaps of the gaps.
Same number added
a_n = a_1 + (n - 1)d
d is the equal gap. Use it when the first row of gaps is constant.
Number series
Same gap (AP)
a_n = a_1 + (n - 1)d
d is the equal gap from the ladder.
Multiply then adjust
a_{n+1} = k \cdot a_n \pm r
Try k = 2 and 3 first, with r = 1 or 2.
Squares family
a_n = n^2 \pm k \ \text{ or } \ n^2 + n
If the second gaps are all 2, the series is built on n squared.
Same ratio (GP)
a_n = a_1 \cdot r^{\,n-1}
r is the fixed ratio. Test ×2, ×3 and ×1.5.
Letter series
Opposite letter
\text{place of opposite} = 27 - \text{place}
A and Z, B and Y, and so on. Each pair adds up to 27.
Wrap around
\text{place} > 26 \Rightarrow \text{place} - 26,\ \ \text{place} < 1 \Rightarrow \text{place} + 26
Use this whenever the rule takes you past Z or before A.