Write an explicit geometric sequence rule

A sequence is labelled from index 0. How many multiplication steps take it from index 0 to index 3?

Move the index from 0 to 4 for a sequence that starts at 10 and doubles. Compare the index, number of steps and multiplier product

Why does index 0 give the initial value 10?

Replace repeated multiplication with a power

A recurrence uses the previous term. An explicit rule uses only the index and fixed numbers, so it can calculate a distant term directly

1. 1Count from index 0 From the initial value at index 0 to index n, the multiplier is applied n times \(\displaystyle u_n=aR^n\)
2. 2Power only the multiplier The starting value appears once. The multiplier repeats \(\displaystyle u_3=a\times R\times R\times R=aR^3\)

Put the whole fraction inside the power

For a sequence starting at 48, both parts of the fractional multiplier repeat. A decimal such as 0.75 is exactly equivalent here

1. 1Keep a fractional base together The power applies to both numerator and denominator \(\displaystyle \left(\frac34\right)^n=\frac{3^n}{4^n}\)
2. 2Check the starting value Every nonzero base has zeroth power 1, so the rule returns a at index 0 \(\displaystyle u_0=aR^0=a\)

A constant sequence has an explicit rule too

When the multiplier is 1, every power of it is 1. The rule can be written without a power

1. 1Do not multiply by n The constant sequence stays at 9. The expression 9n would change with the index

A sequence starts with 5 at index 0 and triples. Which proposed rule incorrectly repeats the initial value as well as the multiplier?

Find the initial value before writing the rule

If a later term is known, reverse the multiplication steps. Use the given ratio and term to reverse two multiplication steps and find the start

1. 1Recover an unseen start Divide a known later term by the multiplier applied the required number of times \(\displaystyle u_0=\frac{u_k}{R^k}\)
2. 2Then state the full rule The known later value is not automatically the coefficient of R to the power n \(\displaystyle u_n=20\left(\frac32\right)^n\)

Two adjacent indexed terms

For a stated geometric sequence, two adjacent nonzero terms determine the ratio. Use their actual indices to reverse back to index 0

1. 1Find the ratio Divide later by earlier \(\displaystyle R=\frac{81}{54}=\frac32\)
2. 2Recover the start and write the rule The value 54 is two transitions after the start \(\displaystyle \begin{gathered}u_0=\frac{54}{(3/2)^2}=24\\u_n=24\left(\frac32\right)^n\end{gathered}\)