Find term n of a geometric sequence using the rule

A sequence starts at t₀ = 6, then gives t₁ = 12 and t₂ = 24. Predict t₃ if the same multiplier continues

The initial value is t₀ = 12. Move the index and the ratio. Compare a term with the next term, including ratios below 1 and exactly 1

With R held fixed at 1.5, what stays constant as n changes?

From repeated multiplication to a power

A recurrence tells you how to get the next term. A direct rule gives any term from the starting value and the number of multiplications

1. 1Check consecutive ratios For nonzero terms, divide each term by the previous one. A geometric sequence has a constant ratio \(\displaystyle 12/6=24/12=2\)
2. 2State the starting index In this lesson the initial value is t₀. Term tₙ is reached after n multiplications \(\displaystyle t_0=6,\quad t_n=6R^n\)
3. 3Power only the ratio The initial value is used once. The ratio is applied repeatedly \(\displaystyle t_3=6\times2^3=48\)
4. 4Adjust for a first term labelled t₁ From term 1 to term n, count one fewer multiplication than the term number. Neither indexing convention is universal \(\displaystyle t_n=t_1R^{n-1}\)
5. 5Use exact supplied values Keep fractions and full calculator precision through the calculation. Round only when asked

Sequence

\(\displaystyle \begin{gathered}t_0=8\\ t_1=12\\ t_2=18\end{gathered}\)

Find the common ratio

Sequence

\(\displaystyle \begin{gathered}t_0=3\\ t_{n+1}=2t_n\end{gathered}\)

How many multiplications take the sequence from t₀ to t₅?

Sequence

\(\displaystyle \begin{gathered}t_0=3\\ t_{n+1}=2t_n\end{gathered}\)

Find t₅ using that rule

Sequence

\(\displaystyle \begin{gathered}t_0=160\\ t_{n+1}=\tfrac12t_n\end{gathered}\)

Now use a decaying sequence. Find t₄

A new sequence is labelled from t₁ = 3, with each term twice the previous term. Which rule gives t₅?

Sequence

\(\displaystyle \begin{gathered}t_3=54\\ R=3\end{gathered}\)

Recover the initial value from a later term and the ratio

Which positive sequence is geometric?

Sequence

\(\displaystyle \begin{gathered}t_0=80\\ R=1-0.25\end{gathered}\)

A model starts at 80 and loses 25% each period. Find its value after 3 periods

For t₀ > 0 and 0 < R < 1, what does the mathematical model do as n increases?

A population model doubles forever. Why should that long-term prediction be questioned?

Sequence

\(\displaystyle \begin{gathered}a_1=7\\ R=3\end{gathered}\)

Transfer: the first term is labelled a₁ = 7 and each next term is three times the previous one. Find a₄