Write and interpret geometric recurrence rules

Two sequences multiply by 2 each step but start at different positive values. Must they have the same terms?

Keep the initial value at 32. Change the multiplier and compare the indexed rows. Follow how each new value becomes the input to the next step

What stays fixed when only R changes?

Two equations, one sequence

Use u(0) for the starting value and u(n+1) for the term immediately after u(n). Parentheses and subscripts represent the same indexed terms here

1. 1Include the initial equation The rule alone does not identify one sequence. State its value at index 0. \(\displaystyle u_0=a\)
2. 2Use the current term The next term is R times the immediately preceding term, not R times the initial value every time \(\displaystyle u_{n+1}=Ru_n\)

What is missing from a rule that doubles the current term to obtain the next?

Fractional multipliers

A multiplier between 0 and 1 decreases a positive value. For the stated geometric sequence 80,60,45, the multiplier is the fraction shown below. Retain the fraction or an exactly equal decimal

1. 1Check consecutive pairs Both quotients agree \(\displaystyle \frac{60}{80}=\frac{45}{60}=\frac34\)
2. 2Write the complete rule State the index 0 value and the forward multiplier \(\displaystyle u_0=80,\qquad u_{n+1}=\frac34u_n\)

The constant case

A repeated positive value still has a geometric recurrence. Its multiplier is 1. The initial value is still required

1. 1Do not add 1 Multiplication by 1 preserves a term; adding 1 would change it

Iterate using the latest term

Start at index 0 and apply the rule once for each next index. The number of multiplications is not the number of displayed terms

1. 1First application Use the initial value \(\displaystyle u_1=\frac12\times20=10\)
2. 2Second application Use the newly obtained 10, not 20 again \(\displaystyle u_2=\frac12\times10=5\)

Repair a whole rule

Check the initial equation and the multiplier separately against the stated sequence. A correct multiplier does not compensate for a wrong starting value

1. 1Check index 0 Compare the initial equation with the value at index 0
2. 2Check every displayed step Multiplying each supplied term by the proposed R must give the next supplied term

Read symbolic conditions

A parameter such as a or R stands for a number meeting the stated conditions. Keep the positive starting-value condition when describing growth or decay

1. 1Growth Every term remains positive and each next term is larger \(\displaystyle a>0,\quad R>1\)
2. 2Decay Every term remains positive and each next term is smaller \(\displaystyle a>0,\quad 0<R<1\)
3. 3Constant sequence Every term remains equal to a \(\displaystyle a>0,\quad R=1\)