Sequences, Series & Finance SL

IB Mathematics: Applications & Interpretation · Topic 1: Number & Algebra

JMaths

www.jmaths.xyz

1

Arithmetic Sequences & Series

An arithmetic sequence has a common difference $d$ between consecutive terms.

General term (nth term)

$ u_n = u_1 + (n - 1)d $

Sum of $ n $ terms

$ S_n = \frac{n}{2} (2u_1 + (n - 1)d) $

or $ S_{n} = \frac{n}{2}(u_1 + u_{n}) $

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When to use: Regular payments of the same amount, linear growth, equally spaced values (e.g. salary increasing by a fixed amount each year).

Worked Example

The 3rd term of an arithmetic sequence is 14 and the 7th term is 26. Find $u_1$ and $d$, and the sum of the first 20 terms.

$ u_3 = u_1 + 2d = 14 $ & $ u_7 = u_1 + 6d = 26 $

Subtract: $ 4d = 12 $, so $ d = 3 $

Substitute: $ u_1 = 14 - 2(3) = 8 $

$ S_{20} = \frac{20}{2} (2(8) + 19(3)) = 10 \times 73 = 730 $

Answer: $ u_1 = 8 $, $ d = 3 $, $ S_{20} = 730 $

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GDC for sums & sigma: Generate or sum a sequence directly. TI-84: sum(seq(8+(X−1)*3, X, 1, 20)) → 730. Casio: OPTN → LIST → Seq then Sum. Faster and avoids arithmetic slips on long sums.

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Common error: Using $n$ instead of $(n-1)$. The 5th term uses $(n-1)=4$, not 5. Always check: $ u_1 = u_1 + 0 \times d $.

2

Geometric Sequences & Series

A geometric sequence has a common ratio $r$ between consecutive terms. Find $r$ by dividing any term by the previous one.

General term (nth term)

$ u_n = u_1 \times r^{(n - 1)} $

Sum of $ n $ terms

$ S_n = u_1 \times (r^n - 1) / (r - 1) $

$ r \neq 1 $

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When to use: Percentage growth/decay, populations, depreciation, repeated multiplication scenarios. Use the geometric link for compound interest only when an algebraic relationship is asked for — in finance questions, use the TVM solver (§3).

Worked Example

A car bought for \$25 000 depreciates by 15% per year. Find its value after 6 years.

$ u_1 = 25000 $, $ r = 1 - 0.15 = 0.85 $ (depreciation means $r<1$)

$ u_7 = 25000 \times 0.85^6 = 25000 \times 0.37715... = 9428.74 $

Answer: \$9428.74 (note: after 6 years = 7th term, but year 0 is $u_1$, so $n=7$ gives exponent 6)

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Common error: Confusing the exponent. After 6 years of depreciation from the purchase price, raise $r$ to the power 6, not 7. Think: year 0 = purchase, year 6 = sixth year.

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Common error: Writing $ r = 0.15 $ for 15% depreciation. The ratio is the multiplier: $ r = 0.85 $ (you keep 85%). For growth of 5%, $ r = 1.05 $.

Sequences, Series & Finance SL

IB Mathematics: Applications & Interpretation · Topic 1: Number & Algebra

JMaths

www.jmaths.xyz

3

Sigma (Σ) Notation

A sigma expression is just a sum of a sequence. Identify the sequence inside, decide whether it is arithmetic or geometric, then use $S_n$ (or your GDC).

Arithmetic sum

$ \sum_{k=1}^{n} \big(a + (k-1)d\big) = S_n $

Geometric sum

$ \sum_{k=1}^{n} a\,r^{\,k-1} = S_n $

Read it off: the lower number is the start, the upper number is the end, and the expression gives the $k$th term. Substitute $k=1$ to find $u_1$; the change per step is $d$ (arithmetic) or the multiplier is $r$ (geometric).

Worked Example

Evaluate $ \displaystyle\sum_{k=1}^{20} (3k+5) $.

$k=1$: $ u_1 = 3(1)+5 = 8 $; each step adds 3, so $ d = 3 $ (arithmetic), $ n = 20 $.

$ u_{20} = 3(20)+5 = 65 $, so $ S_{20} = \frac{20}{2}(8 + 65) = 10 \times 73 = 730 $

Answer: $ 730 $

TI-84 Plus CE

[2nd][STAT] (LIST) → MATH → sum( , and OPS → seq(
sum(seq(3X+5, X, 1, 20)) → 730

TI-Nspire CX II

On a Calculator page, use the Σ summation template (press the templates key → choose Σ):
enter k=1 (bottom), 20 (top), 3k+5 (body) → 730
(equivalent typed form: sum(seq(3k+5, k, 1, 20)))

Casio fx-CG50

Run-Matrix: OPTN → LIST → Seq to build the list, then
OPTN → LIST → Sum over it → 730

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GDC-first: For any Σ question the safest method is to type it into the calculator — no formula slip. Use the closed form only if asked to "show" or work "without technology".

4

Simple vs Compound Interest

Simple interest is paid only on the original amount each period; compound interest is paid on the running balance (interest on interest). Read the question carefully — recent papers test both.

Simple interest (not in booklet)

$ I = \dfrac{PV \cdot r \cdot n}{100} $

total value $ = PV + I $; $r$ in %, $n$ periods

Compound (booklet / TVM)

$ FV = PV\left(1 + \dfrac{r}{100k}\right)^{kn} $

interest $ = FV - PV $

Worked Example — Compare

\$2000 is invested for 3 years at 5% per year. Find the interest under (a) simple and (b) compound (annual) interest.

(a) Simple: $ I = \dfrac{2000 \times 5 \times 3}{100} = 300 $, so total $ = \$2300 $.

(b) Compound: $ FV = 2000(1.05)^3 = 2315.25 $, interest $ = \$315.25 $.

Answer: simple interest \$300; compound interest \$315.25 (compound earns more).

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Common error: Using the TVM solver for a simple-interest question. The solver only does compound interest — for simple interest use $ I = \dfrac{PV \cdot r \cdot n}{100} $ by hand.

Sequences, Series & Finance SL

IB Mathematics: Applications & Interpretation · Topic 1: Number & Algebra

JMaths

www.jmaths.xyz

5

Compound Interest & TVM Solver

Compound interest formula (given in formula booklet)

$ FV = PV \times (1 + r/100k)^{kn} $

PV = present value, FV = future value, $r$ = annual rate (%), $k$ = compounding periods per year, $n$ = years

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IB Exam tip: The TVM solver is faster and expected for most finance questions. The formula is only needed when explicitly asked to "show" or when the question says "without technology".

Worked Example — Savings

Priya invests \$5000 at 4.2% p.a. compounded monthly. Find the value after 8 years.

N = 96 (= 8 $ \times $ 12 months), I% = 4.2, PV = $-5000$, PMT = 0, FV = ?, P/Y = 12, C/Y = 12

Answer: FV = \$6992.59

TVM Solver Setup by Calculator

TI-84 Plus CE

[APPS] → Finance → TVM Solver
N = 96    (total periods = 8 yr × 12)
I% = 4.2    (annual interest rate)
PV = -5000    (negative = money paid out)
PMT = 0    (no regular payments)
FV = ?    (cursor here, then press [ALPHA][ENTER] to solve)
P/Y = 12    C/Y = 12    (monthly compounding)

TI-Nspire CX II

[Menu] → Finance → Finance Solver
N = 96    I(%) = 4.2    PV = -5000
PMT = 0    FV = (leave blank)
PpY = 12    CpY = 12
Leave the FV field blank, tab to it and press [Enter] to solve.

Casio fx-CG50

[MENU] → Financial → Compound Interest
n = 96 I% = 4.2 PV = -5000
PMT = 0 P/Y = 12 C/Y = 12
Enter the known values, then press the F-key under FV (F5) to solve. There is no generic SOLVE key on this screen.

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Common error: Forgetting the sign convention. Money you pay out (invest) is negative. Money you receive is positive. If $ PV = -5000 $, then FV will be positive.

Sequences, Series & Finance SL

IB Mathematics: Applications & Interpretation · Topic 1: Number & Algebra

JMaths

www.jmaths.xyz

6

Savings Plans & Annuities (PMT ≠ 0)

A very common AI SL pattern: a lump sum and/or regular deposits. Put every cash flow in the TVM solver with the right sign.

Sign map: opening lump sum → PV (negative, money out); regular deposit → PMT (negative); target balance → FV (positive). For a drawdown annuity (taking money out), PV is negative (paid in) and PMT is positive (received). Deposits at the end of each period = END mode (the default).

Worked Example — Regular Savings

Pierre invests \$1500 at the end of each month for 10 years at 3.6% p.a. compounded monthly. Find the value of the plan.

N = 120 (= 10 $ \times $ 12), I% = 3.6, PV = 0, PMT = $-1500$, FV = ?, P/Y = 12, C/Y = 12

Solve for FV (PMT timing = END, the default).

Answer: FV = \$216 278.58

TI-84 Plus CE

[APPS] → Finance → TVM Solver
N = 120    I% = 3.6    PV = 0
PMT = -1500    FV = ? (cursor here)
P/Y = 12    C/Y = 12    PMT: END
Press [ALPHA][ENTER] on FV.

TI-Nspire CX II

[Menu] → Finance → Finance Solver
N = 120    I(%) = 3.6    PV = 0
PMT = -1500    FV = (leave blank)
PpY = 12    CpY = 12    PmtAt: END
Leave FV blank, tab to it, press [Enter].

Casio fx-CG50

[MENU] → Financial → Compound Interest
n = 120 I% = 3.6 PV = 0
PMT = -1500 P/Y = 12 C/Y = 12
Set payment timing to End, then press the F-key under FV (F5).

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Drawdown / "how long does it last?": a retiree with PV = $-300000$, withdrawing PMT = $+2800$/month at I% = 3.8, FV = 0 — leave N blank and solve to find how many months the fund lasts.

7

Solving for the Interest Rate (I%)

Recent papers ask you to find the rate. Enter every other field with the correct sign, leave I% blank, and solve.

Worked Example — Find the Rate

\$10 000 is invested and \$800 is added at the end of each month. After 10 years (monthly compounding) the balance is \$160 000. Find the annual interest rate.

N = 120, PV = $-10000$, PMT = $-800$, FV = 160000, P/Y = 12, C/Y = 12

Leave I% blank and solve.

Answer: $ I\% \approx 6.40 $ (3 s.f.) per year

TI-84 Plus CE

[APPS] → Finance → TVM Solver
N = 120    I% = ? (cursor here)
PV = -10000    PMT = -800    FV = 160000
P/Y = 12    C/Y = 12
Press [ALPHA][ENTER] on I%.

TI-Nspire CX II

[Menu] → Finance → Finance Solver
N = 120    I(%) = (leave blank)
PV = -10000    PMT = -800    FV = 160000
PpY = 12    CpY = 12
Leave I(%) blank, tab to it, press [Enter].

Casio fx-CG50

[MENU] → Financial → Compound Interest
n = 120 PV = -10000 PMT = -800
FV = 160000 P/Y = 12 C/Y = 12
Press the F-key under I% (F2) to solve.

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Common error: Sign clashes. If PV and PMT are money paid in, both must be negative and FV positive, or the solver returns an error / a nonsense rate.

Sequences, Series & Finance SL

IB Mathematics: Applications & Interpretation · Topic 1: Number & Algebra

JMaths

www.jmaths.xyz

8

Loans (TVM with PMT)

For loans and savings with regular payments, use the TVM solver with a non-zero PMT value.

Worked Example — Loan Repayment

Marco borrows \$12 000 at 6.5% p.a. compounded monthly. He repays the loan with equal monthly payments over 5 years. Find the monthly payment.

N = 60 (= 5 $ \times $ 12 months), I% = 6.5, PV = 12000 (received), PMT = ?, FV = 0 (fully repaid)

P/Y = 12, C/Y = 12

Solve for PMT

Answer: PMT = −\$234.79 per month (negative = money paid out)

TI-84 Plus CE — Loan

[APPS] → Finance → TVM Solver
N = 60    I% = 6.5    PV = 12000
PMT = ?    (cursor here, [ALPHA][ENTER])
FV = 0    P/Y = 12    C/Y = 12

TI-Nspire CX II — Loan

[Menu] → Finance → Finance Solver
N = 60 I(%) = 6.5 PV = 12000
PMT = (leave blank) FV = 0 PpY = 12 CpY = 12
Leave the PMT field blank, tab to it and press [Enter].

Casio fx-CG50 — Loan

[MENU] → Financial → Compound Interest
n = 60 I% = 6.5 PV = 12000
FV = 0 P/Y = 12 C/Y = 12
Enter the known values, then press the F-key under PMT (F4) to solve. There is no generic SOLVE key on this screen.

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Total paid vs total interest: Total paid = |PMT| $ \times $ N = $ 234.79 \times 60 = $ \$14 087.40. Interest = \$14 087.40 − \$12 000 = \$2087.40.

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Exam Traps & Key Reminders

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N is not always years. If P/Y = 12, then N = total number of months. 5 years $ \to $ N = 60.

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P/Y = C/Y in IB. In IB AI exams, always set P/Y = C/Y (both equal the compounding frequency, e.g. 12 for monthly). This keeps calculations straightforward.

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"Per annum" means per year. I% is always the annual rate in the TVM solver. Never divide I% yourself — the calculator handles it using P/Y and C/Y.

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Rounding too early. Only round your final answer (usually 2 d.p. for money). Keep full precision in intermediate steps.

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Arithmetic vs Geometric. Check the question context: "increases by \$50 each year" = arithmetic. "Increases by 5% each year" = geometric.

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Formula booklet: Both the arithmetic and geometric formulae (u_n and S_n) are given. The compound interest formula FV = PV(1 + r/100k)^kn is also given. You do NOT need to memorise them — but know which one to pick.

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3 sf rule: Unless the question specifies otherwise, give answers correct to 3 significant figures. For money, give to 2 decimal places (nearest cent).