Arithmetic Sequences I
IB Mathematics AA · SL & HL · Shadow Worksheet
Practice
Question 1
Consider the sequence: \(5, 9, 13, 17, \ldots\)
(a)Find the \(n\)th term.
(b)Find the 40th term.
(c)Is 201 in the sequence?
(a)
First term \(a=5\), common difference \(d=4\)
(c)
Solve \(4n+1=201\)
\(n=50\), integer ✓
Answer
Yes, it is the 50th term.
Question 2
An arithmetic sequence has first term 8 and common difference \(-3\).
(a)Write an expression for the \(n\)th term \(u_n\).
(b)Find how many terms are greater than \(-100\).
(b)
Solve \(11-3n>-100\)
\(n<37\)
Question 3
We have the sequence \(u_n = 5n - 3\).
(a)Find \(u_{n+1}\) in terms of \(n\).
(b)Hence show that the sequence is arithmetic.
(b)
Common difference
\(u_{n+1}-u_n=(5n+2)-(5n-3)=5\), constant ✓
Question 4
Find \(k\), given that the following sequences are arithmetic:
(a)\(3, k, 11\)
(b)\(4k-3,\; 5k+2,\; 7k+1\)
(c)\(k^2-1,\; 2k+1,\; 4k-1\)
(b)
Equal differences
\((5k+2)-(4k-3)=(7k+1)-(5k+2)\Rightarrow k+5=2k-1\Rightarrow k=6\)
Answer
\(k=6\) (terms: \(21,32,43\), common difference \(11\))
(c)
Equal differences
\((2k+1)-(k^2-1)=(4k-1)-(2k+1)\Rightarrow 2k-k^2+2=2k-2\Rightarrow k^2=4\)
Answer
\(k=2\) (taking positive value)
Question 5
For the following arithmetic sequences find \(u_n\):
(a)\(u_4=19\) and \(u_{10}=43\)
(b)\(u_3=-8\) and \(u_{12}=-35\)
(a)
6 steps of \(d\)
\(6d=24\Rightarrow d=4\); \(a=u_4-3d=7\)
(b)
9 steps of \(d\)
\(9d=-27\Rightarrow d=-3\); \(a=u_3-2d=-2\)
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