Algebra · Sequences
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The nth term of a linear sequence
Why a constant difference makes a sequence "dn + something", how the zeroth term pins down the constant, and how the nth term answers "is 122 in this sequence?" with an equation instead of a hunch.
Algebra · Sequences
The nth term of a linear sequence
Why a constant difference makes a sequence "dn + something", how the zeroth term pins down the constant, and how the nth term answers "is 122 in this sequence?" with an equation instead of a hunch.
Why it works
A linear (arithmetic) sequence climbs by the same amount each step: adds 4 every time. Two different kinds of rule describe it:- the term-to-term rule — "add 4" — tells you how to get the NEXT term;
- the nth term — a formula in — hands you ANY term directly.
Why ? Because the sequence is the times table, shifted. The 4× table is ; our sequence sits exactly above it. So the nth term is
The quick way to the shift: step BACK once from the first term — the "zeroth term" is , and that's the constant. (Using the first term as the constant, , is the classic slip — its first term would be 11.) Check by substituting : . ✓
Descending sequences have a negative . subtracts 4, so the nth term is . Check: gives 30. ✓
"Is 122 a term?" is an equation, not a feeling. For (nth term ): set , so and . Positions are whole numbers — there is no 24.8th term — so 122 is NOT in the sequence, and that little calculation IS the required "explain why".
Building the rule from further-apart clues. If the 3rd term is 13 and the 7th is 29, the sequence climbed in steps, so ; step back from the 3rd term to the zeroth: . The nth term is — check both givens. ✓