Module 5 of 46Lesson 1 of 11 · Sequences 1 · Arithmetic sequences: the nth term
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Sequences, series & financial applications · Essential understanding 1

Arithmetic sequences: the nth term

IB AA SL 1.2with Prof. Kitso
💻 Using the TI-Nspire CX Student Software on a laptop
  • The screens in these notes are drawn exactly as the Student Software shows them. The keypad on the left of the software works like the handheld — click its keys with the mouse.
  • menu: click menu on the on-screen keypad (or Documents Toolbox ▸ 🔧). The menu numbers and letters are the same as in these notes.
  • enter = Enter  ·  tab = Tab (next box)  ·  esc = Esc  ·  del = Backspace.
  • Powers: type ^ (Shift+6), the power, then press → to come back down before typing more. x² = ^2 then →.
  • Fraction: type /  ·  √: type sqrt(  ·  π: type pi  ·  eˣ: type exp(  ·  store: type :=. For EE (×10ⁿ) and i, click them on the keypad.
  • New page: Ctrl+I (or the Insert menu)  ·  change page: Ctrl+← / Ctrl+→  ·  graph entry line: Ctrl+G  ·  var: Ctrl+L.
  • Radian / Degree: click Settings in the status bar at the bottom of the window. The mode (RAD or DEG) shows in the top bar.
  • IB exam mode (Press-to-Test): everything in these notes works on the non-CAS TI-Nspire with the IB exam settings. In exam mode there are no dot-product, cross-product, unit-vector or eigenvalue keys and no Diff Eq graphs — the notes show the allowed way (type the components, use polyRoots on the characteristic equation, do Euler steps on a Calculator page).
  • If you type nDeriv(, the software shows it as centralDiff( — it is the same command.
The big idea: An arithmetic sequence is a list of numbers where you add the same amount every time. One short formula gives you any term without writing out the whole list.
By the end of this lesson you can:
  • recognise an arithmetic sequence and find its common difference d
  • use to find any term
  • find out whether a number is a term of a sequence, and which one
  • find and when two terms are given
  • use the GDC to list terms with seq(
💡 AA Paper 1 has no calculator
In AA, Paper 1 is calculator-free, so the by-hand working is what earns the marks there. Use the TI-Nspire steps for Paper 2, and to check your hand answers.
🧑🏾 Imagine this…
Thato starts a savings jar with P20. Every week she adds P5, so the jar holds P20, P25, P30, P35 … Each step is “plus 5”. That is an arithmetic sequence. Now: how much is in the jar in week 40 (week 1 is the P20)? You could add 5 again and again, or you could notice the pattern: start + (number of jumps) × 5. Week 40 is 39 jumps after week 1, so 20 + 39 × 5 = P215. The formula just writes this idea down.
Sequence
an ordered list of numbers (called terms), for example 3, 8, 13, 18, …
the first term, the second term and the nth term. The small number is the position.
Arithmetic sequence
each term is the one before plus a fixed number.
Common difference
the fixed number that is added: . It can be negative (the sequence goes down) or a decimal.
★ The nth term of an arithmetic sequence
  •   (in the IB formula booklet)
  • Why ? To reach term from term 1 you make jumps, not jumps. To get to the 5th term you jump 4 times.
  • To test for an arithmetic sequence, check that the difference between neighbours is always the same.
  • The formula can be tidied to : a straight-line pattern with gradient .
⚠️ Watch out
  • Use , not . A common slip is , which gives the term after the one you wanted.
  • Always do second term minus first term, never the other way round. For 20, 17, 14, … the difference is , not .
  • A sequence is not arithmetic just because it increases. 2, 4, 8, 16 grows, but the jumps (2, 4, 8) are different.
💡 Command terms
Write down = no working needed. Find = show the formula and substitute. Show that = you must reach the printed answer with every step visible. If the question says the nth term, the answer is an expression in n (for example ), not a number.
🧪 Try it yourselfBuild an arithmetic sequence
This starts with the sequence from Example 1.2 (first term 7, d = 5). Change u₁ and d and watch the bars. Try a negative d. Check u₂₀ against the formula.

Worked examples

Example 1.1Spotting an arithmetic sequence
Which of these are arithmetic? For each one that is, write down d.
(i) 4, 9, 14, 19, …   (ii) 50, 44, 38, 32, …   (iii) 2, 4, 8, 16, …   (iv) 1.5, 2.0, 2.5, 3.0, …
🧠 Think it through
  1. Subtract each term from the next one.
  2. (i) 9 − 4 = 5, 14 − 9 = 5, 19 − 14 = 5: the same every time.
  3. (ii) 44 − 50 = −6, 38 − 44 = −6, 32 − 38 = −6: the same.
  4. (iii) 4 − 2 = 2 but 8 − 4 = 4: not the same.
  5. (iv) 0.5 every time.
🔢 On your TI-Nspire CX II
  1. You can let the GDC do the subtracting. On a Calculator page type the terms as a list, then use deltaList( to get the differences.
    1.1*DocRADdeltaList({4,9,14,19}){5,5,5}deltaList({2,4,8,16}){2,4,8}
✅ Answer (i) arithmetic, d = 5. (ii) arithmetic, d = −6. (iii) not arithmetic (differences 2, 4, 8). (iv) arithmetic, d = 0.5.
In an exam you only need to show two or three differences; one matching pair is not enough to prove it.
Example 1.2Finding a term far along
A sequence begins 7, 12, 17, 22, … Find (a) the common difference, (b) the 20th term, (c) a formula for un.
🧠 Think it through
  1. , and .
  2. For the 20th term, : there are 19 jumps of 5.
  3. General term: , then tidy up.
✍️ By hand
🔢 On your TI-Nspire CX II
  1. On the Calculator page type seq(7+(k-1)*5,k,1,6) and press enter ↵💻 Enter to list the first 6 terms (a good check on your formula).
    📍 Where to find it on the TI-Nspire
    seq( is in menu💻 click menu on keypad▸6: Statistics▸4: List Operations▸5: Sequence
    1.1*DocRADseq(7+(k−1)·5,k,1,6){7,12,17,22,27,32}
  2. For the 20th term, type the formula directly.
    1.1*DocRADseq(7+(k−1)·5,k,1,6){7,12,17,22,27,32}7+(20−1)·5102
✅ Answer (a) d = 5. (b) u₂₀ = 102. (c) u_n = 5n + 2.
Check: put n = 1 into 5n + 2. You get 7, the first term. Always test your formula on a term you know.
Example 1.3Is it a term? Which one?
The sequence 3, 10, 17, 24, … continues in the same way. (a) Is 311 a term of the sequence? (b) Is 250 a term?
🧠 Think it through
  1. and , so .
  2. Set the formula equal to the number and solve for n.
  3. A term number must be a whole number (1, 2, 3, …). If n is a fraction, the number is not in the sequence.
✍️ By hand
(a)
(b)
🔢 On your TI-Nspire CX II
  1. Go to menu💻 click menu on keypad▸3: Algebra▸1: Numerical Solve and type the equation.
    1.1*DocRADnSolve(3+7·(n−1)=311,n)45.nSolve(3+7·(n−1)=250,n)36.2857
✅ Answer (a) Yes, 311 is the 45th term. (b) No: n = 36.29… is not a whole number.
Quick shortcut: every term is 3 more than a multiple of 7 (3, 10, 17 …). 311 − 3 = 308 = 7 × 44 works; 250 − 3 = 247 is not a multiple of 7.
Example 1.4Finding u₁ and d from two terms
In an arithmetic sequence the 4th term is 17 and the 9th term is 42. Find the first term and the common difference, and hence the 30th term.
🧠 Think it through
  1. Write each fact with the formula: and .
  2. You now have two equations and two unknowns.
  3. Subtract the first from the second: the u₁ cancels and leaves 5d = 25.
✍️ By hand
(2) − (1):
Into (1):
🔢 On your TI-Nspire CX II
  1. Use the simultaneous-equation solver: type the equations with the unknown names a (for u₁) and d.
    1.1*DocRADlinSolvea+3·d=17a+8·d=42,{a,d}{2,5}2+29·5147
  2. Make sure you can still do it by hand, since marks are given for the method on Paper 1.
✅ Answer u₁ = 2, d = 5, u₃₀ = 147.
Shortcut: from the 4th to the 9th term there are 9 − 4 = 5 jumps, and the terms rose by 42 − 17 = 25, so .
Example 1.5First term to go past a limit
Neo has P85 in a savings account in month 1. Each month her balance grows by P15, so the balances are 85, 100, 115, … In which month does her balance first exceed P500?
🧠 Think it through
  1. , , so .
  2. “Exceed” means greater than: solve .
  3. Solve as if it were an equation, then round n UP to the next whole number, because month 28.67 does not exist.
  4. Check both sides: the previous month must be under 500 and your month over 500.
✍️ By hand
So the smallest whole n is 29. Check the months either side:
(not enough)
(yes)
🔢 On your TI-Nspire CX II
  1. Make a table. On a Lists & Spreadsheet page name column A n with the numbers 25 to 31 and column B u, with formula =85+15*(n-1) in its grey formula cell.
    1.1*DocRAD◆An25262728293031Bu=85+15*(n-1)445460475490505520535CD1234567A125
  2. Find the first row where u is over 500: n = 29.
  3. Alternatively solve the equation and round up.
    1.1*DocRADnSolve(85+15·(n−1)=500,n)28.6667
✅ Answer Month 29 (balance P505).
Rounding rule: for “first term greater than”, round n up, never to the nearest. If the solver had said 28.2, the answer would still be month 29.

Challenge questions

⭐ Challenge 1.6Two terms, then a far termPaper 1 style
An arithmetic sequence has u₃ = 11 and u₈ = −4. Find u₁, d and u₂₀.
✏️ Try it on paper and on your calculator first. Then check.
Show the full solution
🧠 Think it through
  1. From the 3rd to the 8th term there are 5 jumps, and the value fell by 15.
  2. So .
  3. .
  4. .
🔢 On your TI-Nspire CX II
  1. Check with the solver.
    1.1*DocRADlinSolvea+2·d=11a+7·d=-4,{a,d}{17,-3}17+19·(-3)-40
✅ Answer u₁ = 17, d = −3, u₂₀ = −40.
⭐ Challenge 1.7A block staircasePaper 2 style
Kabo builds a staircase wall from blocks. Row 1 (the bottom) has 4 blocks and each row above has 3 more blocks than the row below.
  1. (a) Write down the number of blocks in rows 2 and 3.
  2. (b) Find a formula for the number of blocks in row n.
  3. (c) Is there a row with exactly 100 blocks? Is there a row with exactly 150?
  4. (d) Which is the first row with more than 200 blocks?
✏️ Try it on paper and on your calculator first. Then check.
Show the full solution
🧠 Think it through
  1. Row 2: . Row 3: . So , .
  2. .
  3. Solve 3n + 1 = 100, then 3n + 1 = 150. Does n come out whole?
  4. Solve 3n + 1 > 200, then round up.
✍️ By hand
(c) : yes.
: no.
(d)
So . Check: , .
🔢 On your TI-Nspire CX II
  1. Check parts (c) and (d).
    1.1*DocRADnSolve(3·n+1=100,n)33.nSolve(3·n+1=150,n)49.66673·67+12023·66+1199
✅ Answer (a) 7 and 10. (b) u_n = 3n + 1. (c) Yes, row 33 has 100 blocks; no row has 150 (n = 49.67). (d) Row 67 (202 blocks).
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