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Sequences, Variation and Financial Maths

Mathematics · WAEC and JAMB · SS2 and SS3

This is where mathematics meets real money, and WASSCE sets it in naira: interest, discount, hire purchase, VAT and profit. Sequence questions are formula-driven and quick marks once the two standard forms are memorised.

What you need to know

  • In an arithmetic progression each term is found by ADDING a fixed common difference d, so the nth term is a + (n - 1)d, where a is the first term.
  • The common difference is found by subtracting any term from the one after it, not before it: d = second term minus first term. A negative d means the sequence is falling.
  • The sum of the first n terms of an AP is n/2 [2a + (n - 1)d], which can also be written n/2 (first term + last term) when you already know the last term.
  • In a geometric progression each term is found by MULTIPLYING by a fixed common ratio r, so the nth term is a r^(n-1) and r = any term divided by the one before it.
  • A geometric progression only has a sum to infinity when r lies strictly between -1 and 1, and that sum is a/(1 - r). If r is 2, no such sum exists.
  • Direct variation means y = kx, inverse variation means y = k/x, joint variation means y = kxz, and partial variation means y = k + mx. The first step in every variation question is to replace 'varies as' with the correct form and a constant k.
  • Find k using the pair of values the question gives, write the full equation with k substituted, and only then answer the part being asked. Skipping to the answer without finding k loses method marks.
  • Simple interest is I = PRT/100 and it is calculated on the original principal every year, so it grows by the same amount each year.
  • Compound interest adds the interest to the principal before the next year, so the amount after T years is A = P(1 + R/100)^T and the interest is A - P.
  • Percentage profit or loss is always worked on the COST price: profit percent = (selling price - cost price)/cost price x 100.
  • Discount is taken off the marked price, so an item marked N12,000 with a 15 percent discount sells for 0.85 x 12,000 = N10,200.
  • VAT in Nigeria is added to the price the seller charges, so a service priced at N80,000 plus 7.5 percent VAT costs the customer 80,000 x 1.075 = N86,000.
  • In hire purchase the buyer pays a deposit plus instalments, and the total paid always exceeds the cash price. The extra amount is the hire purchase charge, usually expressed as a percentage of the cash price.
  • For depreciation, use the compound formula with a MINUS sign: value after T years = P(1 - R/100)^T. A generator bought at N500,000 depreciating at 10 percent is worth 500,000 x 0.9 = N450,000 after one year.

Key terms

Arithmetic progression
A sequence in which each term after the first is obtained by adding a fixed number called the common difference.
Geometric progression
A sequence in which each term after the first is obtained by multiplying the previous term by a fixed number called the common ratio.
Partial variation
A relationship of the form y = k + mx in which one part is constant and the other varies with x.
Simple interest
Interest calculated only on the original principal for the whole period.
Compound interest
Interest calculated on the principal together with all interest already earned.
Hire purchase
A buying arrangement in which the buyer pays a deposit and the balance in instalments, paying more in total than the cash price.
Depreciation
The fall in the value of an asset over time, usually expressed as a fixed percentage of its value each year.

Formulae

  • AP nth term: Un = a + (n - 1)d
  • AP sum: Sn = n/2 [2a + (n - 1)d]
  • GP nth term: Un = a * r^(n - 1)
  • GP sum: Sn = a(r^n - 1)/(r - 1) for r greater than 1
  • GP sum to infinity: S = a/(1 - r), valid only when -1 < r < 1
  • direct variation: y = kx; inverse variation: y = k/x; joint: y = kxz; partial: y = k + mx
  • simple interest: I = P*R*T/100
  • compound amount: A = P(1 + R/100)^T
  • depreciated value: A = P(1 - R/100)^T
  • percentage profit = (selling price - cost price)/cost price x 100
  • discount price = marked price x (100 - discount percent)/100
  • price with VAT = price x (100 + VAT percent)/100

Worked examples

The fifth term of an arithmetic progression is 11 and the ninth term is 23. Find the first term, the common difference, and the sum of the first 20 terms.

  1. Write both terms with the formula: a + 4d = 11 and a + 8d = 23.
  2. Subtract the first from the second: 4d = 12, so d = 3.
  3. Substitute into a + 4d = 11: a + 12 = 11, so a = -1.
  4. Use the sum formula: S20 = 20/2 [2(-1) + (20 - 1)(3)].
  5. Inside the bracket: -2 + 57 = 55.
  6. So S20 = 10 x 55 = 550.
  7. Check the fifth term: -1 + 4(3) = 11. Correct.

Mr Adeyemi invests N250,000 at 8 percent per annum compound interest for 3 years. Find the amount and the interest earned.

  1. Use A = P(1 + R/100)^T with P = 250,000, R = 8 and T = 3.
  2. Year 1: 250,000 x 1.08 = N270,000.
  3. Year 2: 270,000 x 1.08 = N291,600.
  4. Year 3: 291,600 x 1.08 = N314,928.
  5. So the amount is N314,928.
  6. Interest = amount minus principal = 314,928 - 250,000 = N64,928.
  7. Compare with simple interest, which would be 250,000 x 8 x 3/100 = N60,000, confirming compound interest is larger.

A laptop is sold for N180,000 cash or on hire purchase for a deposit of N40,000 plus twelve monthly instalments of N14,000. Find the hire purchase charge as a percentage of the cash price, correct to one decimal place.

  1. Total of instalments = 12 x 14,000 = N168,000.
  2. Total hire purchase price = deposit + instalments = 40,000 + 168,000 = N208,000.
  3. Hire purchase charge = 208,000 - 180,000 = N28,000.
  4. Express as a percentage of the CASH price: 28,000/180,000 x 100.
  5. 28,000/180,000 = 0.15555..., and multiplying by 100 gives 15.555...
  6. To one decimal place this is 15.6 percent.

The mistake to avoid

Candidates use the simple interest formula when the question says compound, and the difference shows up immediately in the marking. In profit questions, the second common error is dividing the profit by the selling price instead of the cost price, which always gives a smaller, wrong percentage.

In the exam

Write the formula in full before substituting, because the formula itself usually carries a mark even if your arithmetic later slips. For compound interest over three or four years, multiplying year by year is safer than using a power on a calculator and it shows your working. Keep the naira sign and round money to the nearest naira or kobo as the question directs.