Generate the terms of a sequence from a starting value and a rule

For t₀ = 4, followed by adding 3 each time, what is t₁?

Change the starting value and the additive step separately. Predict which points move when you change each control, then check the displayed terms

Which statement is correct?

Read the recurrence one part at a time

The initial value starts the process. A recurrence tells you how to get the next term from the current term. Increase the index by one for each application

1. 1Initial condition t₀ = 7 means the starting term is 7, before any application
2. 2Repeated rule tₙ₊₁ = tₙ − 2 means subtract 2 from the latest term
3. 3Count carefully The first five terms are indexed 0, 1, 2, 3 and 4. Reaching index 4 takes four applications

Displayed terms or rule

\(\displaystyle t_0=7,\quad t_{n+1}=t_n-2\)

Start at t₀ = 7 and subtract 2 each time. Find t₁

Displayed terms or rule

\(\displaystyle t_1=5,\quad t_{n+1}=t_n-2\)

For the same rule, t₁ = 5. Find t₂

Which value should be used to calculate t₃ from the rule tₙ₊₁ = tₙ − 2?

Displayed terms or rule

\(\displaystyle u_0=3,\quad u_{n+1}=1.5u_n\)

Start at u₀ = 3 and multiply by 1.5 each time. Find u₁

Displayed terms or rule

\(\displaystyle u_1=4.5,\quad u_{n+1}=1.5u_n\)

The previous question gives u₁ = 4.5. Find u₂ using the same multiplier

Start at 3 and double repeatedly. Move the application counter from 0 to 6. Compare the current index with the number of terms including the initial value

When n = 4, how many applications and how many terms including t₀ are involved?

Displayed terms or rule

\(\displaystyle v_3=10,\quad v_{n+1}=v_n-4\)

You are given v₃ = 10. The rule subtracts 4 each step. Find v₅

You are given v₃ and asked for v₅. How many further applications are needed?

A sequence starts at 5 and adds 2 each time. Which recurrence matches those instructions?

A sequence starts at 12 and multiplies by one half each time. Which list gives t₀, t₁, t₂ and t₃?

Check an indexed list

Keep a table with an index beside every value. Check both the initial condition and every transition, especially for a negative step or fractional multiplier

1. 1Given versus calculated Copying a supplied starting value is not a new application of the rule
2. 2Later windows If a later term is supplied, count only the steps from its index to the requested index
3. 3Explicit formulas come later A recurrence uses the previous term. A direct nth-term formula uses the index itself; do not confuse the two