A rule that jumps straight there
To reach term 100, must you add 3 ninety-nine times?
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Step 1 of 4
- Term 1 is 5. Term 10 is 32.
- The walker takes 9 hops of +3, one after another.
- The jumper works out 5 + 9 × 3 in one line.
- Same answer. One working instead of nine.
The rule
No. One working with the position gives term 100.
An explicit rule uses the position n to give the term. It needs no earlier term.
The position n is a term's place: 1st, 2nd, 3rd.
For 5, 8, 11, … the rule is tₙ = 3n + 2. Check: n = 1 gives 3 × 1 + 2 = 5.
Use n = 10: t₁₀ = 3 × 10 + 2 = 32. One working.
Use n = 100: t₁₀₀ = 3 × 100 + 2 = 302. Still one working.
Which position is a number?
Set the rule equal to the number. Then find n.
For 137: 3n + 2 = 137. Take 2 from both sides: 3n = 135. Divide by 3: n = 45.
For 138: 3n + 2 = 138, so 3n = 136. Then n is about 45.33. That is not a whole number (1, 2, 3, …). So 138 is not a term.
A position must be a whole number. A list has no 45.33rd term.