Arithmetic sequence
- Each term is the one before plus the same number, the common difference \(d\).
- \(u_n = u_1 + (n-1)d\)
- From the 1st term to the \(n\)th there are \(n - 1\) steps, not \(n\).
12.1.1
Arithmetic Sequences
12.1.1
Arithmetic Sequences
By the end of this section
- Find any term of an arithmetic sequence
- Find the first term and the common difference from two terms
- Find which term has a given value
TI-Nspire: New Document (not Scratchpad). Menu > Algebra > Solve System… solves the two equations for the first term and the common difference.
Key points
Two terms given
- Write each term as an equation in \(u_1\) and \(d\), then solve the pair together.
On the calculator
Solve two equations
New Document (not Scratchpad) · Menu > Algebra > Solve System…
- Number of equations: 2. Variables: x, y.
- Type \(u_1\) as x and \(d\) as y.
Solve for n
nSolve(equation, n)
- Type the whole equation, with its equals sign: nSolve(1 + 3(n − 1) = 100, n).
Screenshot and video to come
Explore: arithmetic and geometric sequences
Choose arithmetic or geometric, then change the first term and d or r. Before you tick the box, predict what the sums Sₙ will look like.
| n | uₙ | Sₙ |
|---|
Move a slider to change the sequence.
- Arithmetic: what shape do the dots make, and what does d do to it? What happens when d is negative?
- Geometric: set r between 0 and 1, then below 0, then above 1. Describe the dots each time.
- Tick the sums. For which settings do the sums Sₙ climb in a straight line, and for which do they level off?
- Set r = 1 in the geometric sequence. What is Sₙ, and why can the formula not be used?
Worked examples
Problems like the ones in class. Pick one, try it, then open the working.
AI SL Sequences and Series slides · Review 1
Find a term
An arithmetic sequence has first term \(u_1 = 7\) and common difference \(d = 4\). Find \(u_{20}\).
Working
Have a go first. Then open the working one line at a time.
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The rule
\(u_n = u_1 + (n-1)d\)
The first term, then \(n - 1\) steps of \(d\).
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The 20th term
\(u_{20} = 7 + (20 - 1)(4) = 83\)
19 steps of 4 from 7.
AI SL Sequences and Series slides · Practice 1a
A term further on
An arithmetic sequence has \(u_1 = 12\) and \(d = 5\). Find \(u_{18}\).
Working
Have a go first. Then open the working one line at a time.
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The 18th term
\(u_{18} = 12 + (18 - 1)(5) = 97\)
17 steps of 5 from 12.
AI SL Sequences and Series slides · Practice 1b
A sequence that goes down
Find the 25th term of 40, 34, 28, 22, …
Working
Have a go first. Then open the working one line at a time.
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First term and difference
\(u_1 = 40\), \(d = 34 - 40 = -6\)
The terms go down by 6.
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The 25th term
\(u_{25} = 40 + (25 - 1)(-6) = -104\)
24 steps of −6 from 40.
AI SL Sequences and Series slides · Practice 1c
Write the rule
For the sequence 4, 11, 18, …, write an expression for \(u_n\).
Working
Have a go first. Then open the working one line at a time.
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First term and difference
\(u_1 = 4\), \(d = 7\)
The terms go up by 7.
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The rule
\(u_n = 4 + (n - 1)(7) = 7n - 3\)
Check: \(n = 1\) gives 4.
AI SL Sequences and Series slides · Practice 1d
Which term is it?
Which term of the sequence 6, 13, 20, … is equal to 209?
Working
Have a go first. Then open the working one line at a time.
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The rule
\(u_n = 6 + (n - 1)(7)\)
\(u_1 = 6\), \(d = 7\).
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Solve for n
\(6 + 7(n - 1) = 209\) gives \(n = 30\)
By hand, or with nSolve on the calculator. It is the 30th term.
A pattern of shapes
Matchsticks are used to make a row of squares. One square uses 4 matchsticks, two squares use 7 and three squares use 10. (a) How many matchsticks make a row of 20 squares? (b) A row uses 154 matchsticks. How many squares are in it?
Working
Have a go first. Then open the working one line at a time.
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Spot the sequence
\(4,\ 7,\ 10,\ \ldots\)
The same amount, 3, is added each time: arithmetic with \(u_1 = 4\) and \(d = 3\).
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The rule
\(u_n = 4 + (n - 1)(3) = 3n + 1\)
Check it on the first term: 3(1) + 1 = 4.
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(a)
\(u_{20} = 3(20) + 1 = 61\)
61 matchsticks.
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(b)
\(3n + 1 = 154 \Rightarrow n = 51\)
A row of 51 squares.
Is it a term?
Is 149 a term of the sequence 9, 16, 23, …? Is 150 a term? If a number is a term, say which term it is.
Working
Have a go first. Then open the working one line at a time.
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The rule
\(u_n = 9 + (n - 1)(7) = 7n + 2\)
Here \(u_1 = 9\) and \(d = 7\).
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Try 149
\(7n + 2 = 149 \Rightarrow n = 21\)
A whole number, so 149 is the 21st term.
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Try 150
\(7n + 2 = 150 \Rightarrow n = 21.14\ldots\)
Not a whole number, so 150 is not a term. A term must have a whole-number position.
AI SL Sequences and Series slides · Review 2
Two terms given
The third term of an arithmetic sequence is 10 and the 15th term is 46. Find the first term and the common difference.
Working
Have a go first. Then open the working one line at a time.
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Two equations
\(u_1 + 2d = 10\) and \(u_1 + 14d = 46\)
Write each term with \(u_n = u_1 + (n-1)d\).
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Solve
\(d = 3\), \(u_1 = 4\)
Menu > Algebra > Solve System… on the calculator, or subtract: \(12d = 36\).
AI SL Sequences and Series slides · Practice 2a
Find the first term and the difference
The 4th term is 17 and the 10th term is 47. Find \(u_1\) and \(d\).
Working
Have a go first. Then open the working one line at a time.
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Two equations
\(u_1 + 3d = 17\) and \(u_1 + 9d = 47\)
The 4th term is 3 steps from the first; the 10th is 9 steps.
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Solve
\(d = 5\), \(u_1 = 2\)
Subtracting gives \(6d = 30\).
AI SL Sequences and Series slides · Practice 2b
A difference that is negative
The 2nd term is 30 and the 12th term is −10. Find \(u_1\) and \(d\).
Working
Have a go first. Then open the working one line at a time.
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Two equations
\(u_1 + d = 30\) and \(u_1 + 11d = -10\)
Ten steps take the sequence from 30 down to −10.
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Solve
\(d = -4\), \(u_1 = 34\)
Subtracting gives \(10d = -40\).
AI SL Sequences and Series slides · Practice 2c
Then find another term
The 5th term is 8 and the 20th term is 53. Find the 40th term.
Working
Have a go first. Then open the working one line at a time.
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Two equations
\(u_1 + 4d = 8\) and \(u_1 + 19d = 53\)
Find \(u_1\) and \(d\) first.
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Solve
\(d = 3\), \(u_1 = -4\)
Subtracting gives \(15d = 45\).
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The 40th term
\(u_{40} = -4 + (40 - 1)(3) = 113\)
39 steps of 3 from −4.
AI SL Sequences and Series slides · Practice 2d
The first negative term
An arithmetic sequence has \(u_3 = 9\) and \(u_7 = 29\). Find the first negative term.
Working
Have a go first. Then open the working one line at a time.
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Two equations
\(u_1 + 2d = 9\) and \(u_1 + 6d = 29\)
Find \(u_1\) and \(d\).
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Solve
\(d = 5\), \(u_1 = -1\)
Subtracting gives \(4d = 20\).
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Look at the first term
\(u_1 = -1\)
The sequence goes up, and it starts below zero. So the first term is the only negative term.