Recognise a repeated multiplicative change without using logarithms.
Extended · Syllabus E2.7
Notes and key methods
An exponential sequence multiplies each term by the same ratio r. If its first term is a at index 1, then \(\displaystyle T_{n}\)=arⁿ⁻¹. Compare consecutive ratios rather than consecutive differences.
To find an index when exact small powers are involved, divide out a and recognise the appropriate power of r. Always check that the proposed index is a positive integer. A formula for a sum of a geometric progression is not required here.
Check the conditions
Use the stated units and precision.
Show the method as well as the final answer.
Three worked examples
Example 1 · Exponential sequences
The sequence 6, 24, 96, 384 has a constant positive multiplicative ratio and begins at \(\displaystyle n=1\).
Find an nth-term formula.
Find the first-term coefficient a in \(\displaystyle T_{n}\)=arⁿ⁻¹.
Find the term at \(\displaystyle n=6\).
Find the next term after the four listed terms.
Show the reasoning
(a)
The consecutive ratio is 4 and the term at \(\displaystyle n=1\) is 6.
Answer: \(T_n=6\times 4^{n-1}\)
(b)
Divide consecutive terms to identify the common multiplier, then count multiplications from index 1.
\(6\)
Answer: \(6\).
(c)
Divide consecutive terms to identify the common multiplier, then count multiplications from index 1.
\(6144\)
Answer: \(6144\).
(d)
Divide consecutive terms to identify the common multiplier, then count multiplications from index 1.
\(1536\)
Answer: \(1536\).
Example 2 · Exponential sequences
The sequence 2, 4, 8, 16 has a constant positive multiplicative ratio and begins at \(\displaystyle n=1\).
Find an nth-term formula.
Find the first-term coefficient a in \(\displaystyle T_{n}\)=arⁿ⁻¹.
Find the term at \(\displaystyle n=7\).
Find the next term after the four listed terms.
Show the reasoning
(a)
The consecutive ratio is 2 and the term at \(\displaystyle n=1\) is 2.
Answer: \(T_n=2\times 2^{n-1}\)
(b)
Divide consecutive terms to identify the common multiplier, then count multiplications from index 1.
\(2\)
Answer: \(2\).
(c)
Divide consecutive terms to identify the common multiplier, then count multiplications from index 1.
\(128\)
Answer: \(128\).
(d)
Divide consecutive terms to identify the common multiplier, then count multiplications from index 1.
\(32\)
Answer: \(32\).
Example 3 · Exponential sequences
The sequence 3, 9, 27, 81 has a constant positive multiplicative ratio and begins at \(\displaystyle n=1\).
Find an nth-term formula.
Find the first-term coefficient a in \(\displaystyle T_{n}\)=arⁿ⁻¹.
Find the term at \(\displaystyle n=8\).
Find the next term after the four listed terms.
Show the reasoning
(a)
The consecutive ratio is 3 and the term at \(\displaystyle n=1\) is 3.
Answer: \(T_n=3\times 3^{n-1}\)
(b)
Divide consecutive terms to identify the common multiplier, then count multiplications from index 1.
\(3\)
Answer: \(3\).
(c)
Divide consecutive terms to identify the common multiplier, then count multiplications from index 1.
\(6561\)
Answer: \(6561\).
(d)
Divide consecutive terms to identify the common multiplier, then count multiplications from index 1.
\(243\)
Answer: \(243\).
Common mistakes
Keep intermediate values exact until the final step.