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Sequences, Series & Finance SL

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

JMaths

www.jmaths.xyz

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} $ = ^{$ 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: 4$ d = 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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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. $

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, compound interest (without GDC), populations, depreciation, repeated multiplication scenarios.

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

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 = 8, I% = 4.2, PV = $ -5000, $ PMT = 0, FV = ?, P/Y = 12, C/Y = 12

Answer: FV = $6976.41

TVM Solver Setup by Calculator

TI-84 Plus CE

[APPS] → Finance → TVM Solver
N = 8    (total years)
I% = 4.2    (annual interest rate)
PV = -5000    (negative = money paid out)
PMT = 0    (no regular payments)
FV = 0    (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 = 8    I(%) = 4.2    PV = -5000
PMT = 0    FV = 0
PpY = 12    CpY = 12
Place cursor on FV, press [Enter] to solve.

Casio fx-CG50

[MENU] → Financial → Compound Interest
n = 8    I% = 4.2    PV = -5000
PMT = 0    FV = 0
P/Y = 12    C/Y = 12
Highlight FV, press [SOLVE] (F6).

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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.

Loans & Annuities (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 = $ - $$234.85 per month (negative = money paid out)

TI-84 Plus CE — Loan

[APPS] → Finance → TVM Solver
N = 60    I% = 6.5    PV = 12000
PMT = 0    (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 = 0 FV = 0 PpY = 12 CpY = 12
Cursor on PMT, press [Enter].

Casio fx-CG50 — Loan

[MENU] → Financial → Compound Interest
n = 60 I% = 6.5 PV = 12000
PMT = 0 FV = 0 P/Y = 12 C/Y = 12
Highlight PMT, press [SOLVE] (F6).

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Total paid vs total interest: Total paid = |PMT| $ \times $ N = 234.85 $ \times 60 = $ $14 091. Interest = $14 091 $ - $ $12 000 = $2091.

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).