Write a recurrence relation for a sequence

Two sequences use “add 3 each time” but start at different values. Must their terms be identical?

Match the target 5, 8, 11, 14 by changing the starting value and signed difference. First vary only one control, then check every row

Which pair matches all target rows?

State the start and the forward update

The initial condition gives the first value. The forward rule tells you how to find the next term from the current term

1. 1Initial condition This fixes the starting value \(\displaystyle t_0=5\)
2. 2Next from current Add 3 to the current value to find the next value \(\displaystyle t_{n+1}=t_n+3\)
3. 3Restricted class Here use a constant additive or multiplicative rule that fits all displayed terms. A finite list alone does not prove a unique unrestricted rule

Zero-indexed values

\(\displaystyle -6,-2,2,6\)

The zero-indexed sequence begins −6, −2, 2, 6. What is the initial value?

Values

\(\displaystyle -6,-2,2,6\)

For −6, −2, 2, 6, find the signed common difference

Which complete recurrence generates −6, −2, 2, 6 from index zero?

Values

\(\displaystyle 11,7,3,-1\)

A sequence begins 11, 7, 3, −1. What is its signed common difference?

A sequence decreases by 4 at each step. Which update represents this?

A multiplier can be fractional

For a multiplicative rule, divide each later value by the preceding value. Check more than one pair

1. 1Growth A positive multiplier greater than 1 increases positive terms
2. 2Decay A multiplier between 0 and 1 decreases positive terms
3. 3Equivalent wording Dividing by 2 is multiplying by one half. These are the same sequence rule

Values

\(\displaystyle 24,12,6,3\)

The terms are 24, 12, 6, 3. What constant multiplier links each pair?

Which complete recurrence generates 24, 12, 6, 3, starting at index zero?

Initial value and rule

\(\displaystyle t_0=8,quad t_{n+1}=\frac{3}{2}t_n\)

Using the given initial value and rule, what is the term at index 1?

Which expression uses the current term rather than the index?

The table gives term values at indices 0, 1, 2 and 3. Which pair defines its additive recurrence?

Does one matching transition prove a proposed rule fits the whole displayed list?

Read, build, then check

For a table or discrete graph, identify the index-zero value first. Compare later values with preceding values to find a constant difference or ratio

1. 1Full recurrence Write the initial condition and the update together
2. 2Independent writing On paper, write your full recurrence and use it to generate two terms. Compare them with the source