Grade 11 Math — Functions & Applications · Arithmetic sequences
The off-by-one that guards the door
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The off-by-one that guards the door

tn=a+(n1)dt_n = a + (n - 1)\,d — read aloud: t-n equals a, plus n minus one, times d. Here aa is the first term and dd the common difference. The n1n - 1 is the whole exam: the first term has taken zero steps, so by term nn only n1n - 1 steps have happened. Term 10 of 2, 5, 8, … is 2+9×3=292 + 9 \times 3 = 29 — nine steps, not ten.

One fact, four sockets: solve it for tnt_n, for aa, for dd, or for nn, and it answers four different exam questions. The ladder is the one you know — pick the formula, rearrange for the target, load the givens, then compute. And when the target is nn, the answer must land on a whole number: term 11½ does not exist, and an answer like it means the arithmetic upstream slipped.