Sequences Mixed — Past Paper Style (HL)
IB Mathematics AA · SL & HL · Shadow Worksheet
Practice
Question 1
Find \(k\) such that \(\displaystyle\sum_{r=1}^{k}(2r+1)=120\).
Expand sum
\(\displaystyle\sum_{r=1}^k(2r+1)=2\cdot\dfrac{k(k+1)}{2}+k=k^2+2k=120\Rightarrow k^2+2k-120=0\Rightarrow(k+12)(k-10)=0\)
Question 2
The 3rd term of a geometric sequence is 4 and the 7th term is \(\tfrac{1}{4}\). Find the sum to infinity.
Find \(r\)
\(r^4=\dfrac{1/4}{4}=\tfrac{1}{16}\Rightarrow r=\pm\tfrac{1}{2}\)
Find \(u_1\)
With \(r=\tfrac{1}{2}\): \(u_1\cdot\tfrac{1}{4}=4\Rightarrow u_1=16\). With \(r=-\tfrac{1}{2}\): same \(u_1=16\)
Sum to infinity
\(S_\infty=\dfrac{16}{1-(\pm\tfrac{1}{2})}\)
Answer
\(S_\infty=32\) (if \(r=\tfrac{1}{2}\)) or \(\dfrac{32}{3}\) (if \(r=-\tfrac{1}{2}\))
Question 3
Prove by induction that \(\displaystyle\sum_{r=1}^n r = \dfrac{n(n+1)}{2}\).
Base case
\(n=1\): LHS\(=1\); RHS\(=\tfrac{1\cdot2}{2}=1\) ✓
Inductive step
Assume true for \(n=k\). For \(n=k+1\): \(\displaystyle\sum_{r=1}^{k+1}r=\dfrac{k(k+1)}{2}+(k+1)=\dfrac{k(k+1)+2(k+1)}{2}=\dfrac{(k+1)(k+2)}{2}\) ✓
Conclusion
True by PMI for all \(n\in\mathbb{Z}^+\). ■
Question 4
An arithmetic sequence has first term 1 and common difference 3.
(a)Find the 2nd term of the arithmetic sequence.
(b)A separate, geometric sequence has first term 1 and common ratio \(r=\dfrac{1}{u_2}\), using the 2nd term found in (a). State \(r\) and find the sum to infinity of the geometric sequence.
(b)
Ratio
\(r=\dfrac{1}{4}\); since \(|r|<1\), the sum to infinity exists
Sum to infinity
\(S_\infty=\dfrac{1}{1-\tfrac{1}{4}}=\dfrac{1}{3/4}\)
Answer
\(r=\tfrac{1}{4}\); \(S_\infty=\tfrac{4}{3}\)
Question 5
The sum of an infinite geometric series is twice its first term. Find the common ratio. Hence find the first term if the sum of the first 5 terms is 31.
Condition
\(\dfrac{u_1}{1-r}=2u_1\Rightarrow1-r=\tfrac{1}{2}\Rightarrow r=\tfrac{1}{2}\)
Sum of 5 terms
\(u_1\cdot\dfrac{1-(1/2)^5}{1/2}=2u_1\cdot\dfrac{31}{32}=31\Rightarrow u_1=16\)
Answer
\(r=\tfrac{1}{2},\;u_1=16\)
Generated by Mathski · mathski.io · IB Mathematics AA Shadow Worksheets