Find when a sequence first reaches or passes a given value

Does “first exceeds 14” include a term equal to 14?

Move the index from 0 to 6. Find the first term equal to 14, then the first greater than 14. Notice the difference between reaching and exceeding

For tₙ = 2 + 3n, which index first exceeds 14?

Translate the condition

Use the exact condition before choosing a whole-number index

1. 1Exceeds Use a strict greater-than inequality \(\displaystyle t_n>T\)
2. 2Reaches Equality counts \(\displaystyle t_n\ge T\)
3. 3Falls below Equality does not count \(\displaystyle t_n<T\)

Given values

\(\displaystyle t_n=2+3n\)

For tₙ = 2 + 3n, find the first index reaching 14

Given values

\(\displaystyle t_n=2+3n\)

For tₙ = 2 + 3n, find the first index exceeding 14

Given values

\(\displaystyle t_n=2+3n\)

For tₙ = 2 + 3n, find the first index exceeding 15

An inequality gives n > 4.1. Which is the first valid integer?

Given values

\(\displaystyle u_n=25-4n\)

For uₙ = 25 − 4n, find u₃

Given values

\(\displaystyle u_n=25-4n\)

For the same rule, find u₄

Given values

\(\displaystyle u_n=25-4n\)

Find the first index at which uₙ = 25 − 4n falls below 13

Subtract 25 from both sides of 25 − 4n < 13. Then divide by −4. What follows?

Why check the term immediately before your answer?

Given values

\(\displaystyle t_n=8+2n,\quad n\ge0\)

For tₙ = 8 + 2n, what is the first non-negative index reaching 7?

State the index and justify first

Write the inequality, solve it, choose the smallest permitted integer and check that integer and its predecessor. If the initial term already works, the answer is the initial index