Sequences Mixed (SL)
IB Mathematics AA · SL · Shadow Worksheet
Practice
Question 1
The first three terms of a sequence are \(k-1,\;k+3,\;3k-1\).
(a)Find \(k\) if the sequence is arithmetic.
(b)Find \(k\) if the sequence is geometric.
(a)
Equal differences
\((k+3)-(k-1)=(3k-1)-(k+3)\Rightarrow4=2k-4\)
(b)
Ratio condition
\((k+3)^2=(k-1)(3k-1)\Rightarrow k^2+6k+9=3k^2-4k+1\Rightarrow2k^2-10k-8=0\Rightarrow k^2-5k-4=0\)
Answer
\(k=\dfrac{5\pm\sqrt{41}}{2}\)
Question 2
An arithmetic sequence has \(u_1=a\) and \(u_2=a^2\). Given that \(a>0\), find the value of \(a\) such that the sequence is also geometric.
For geometric
\(\dfrac{u_2}{u_1}=\dfrac{u_3}{u_2}\); common difference in arithmetic: \(d=a^2-a\), so \(u_3=a^2+(a^2-a)=2a^2-a\). Geometric condition: \((a^2)^2=a(2a^2-a)\Rightarrow a^4=2a^3-a^2\Rightarrow a^2(a^2-2a+1)=0\Rightarrow a(a-1)^2=0\)
Answer
\(a=1\) (since \(a>0\))
Question 3
The \(n\)th term of a sequence is given by \(u_n=3^n-2n\).
(a)Write down the first four terms.
(b)Is this sequence arithmetic, geometric, or neither? Justify.
(b)
Check differences and ratios
Differences: 4, 16, 52 — not constant. Ratios: 5, 4.2, 3.48 — not constant.
Question 4
The sum of the first \(n\) terms of a series is \(S_n=2^{n+1}-2\).
(a)Find \(u_1,u_2,u_3\).
(b)Show the series is geometric and state the common ratio.
(a)
\(u_n=S_n-S_{n-1}=2^{n+1}-2^n=2^n\)
Answer
\(u_1=2,u_2=4,u_3=8\)
(b)
Common ratio
\(\dfrac{u_{n+1}}{u_n}=\dfrac{2^{n+1}}{2^n}=2\), constant ✓
Question 5
Three consecutive terms of an arithmetic sequence have sum 24 and product 480. Find the three terms.
Let terms be
\(a-d,\;a,\;a+d\). Sum: \(3a=24\Rightarrow a=8\)
Product
\((8-d)\cdot8\cdot(8+d)=480\Rightarrow8(64-d^2)=480\Rightarrow d^2=4\Rightarrow d=\pm2\)
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