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What does a number in standard form look like?
a × 10ᵏ, with 1 ≤ a < 10 and k a whole number. Example: 4.53 × 10⁴.
In a × 10ᵏ, what is the allowed range for the coefficient a?
1 ≤ a < 10 (at least 1, less than 10). So 7 × 10³ ✓ but 12 × 10³ ✗.
Big number (10 or more): is the exponent positive or negative?
Positive. Example: 52 000 = 5.2 × 10⁴.
Small number (less than 1): is the exponent positive or negative?
Negative. Example: 0.0007 = 7 × 10⁻⁴.
How do you find the exponent k?
Count how many places the point moves to leave one non-zero digit in front. Left → positive, right → negative.
Write 73 000 in standard form.
7.3 × 10⁴.
Your GDC shows 6.1ᴇ-5. What is this in standard form?
6.1 × 10⁻⁵ — the ᴇ symbol means × 10.
Why is 45.3 × 10⁶ not standard form? Fix it.
The coefficient 45.3 is not between 1 and 10. Correct: 4.53 × 10⁷.
When you multiply powers of ten, what do you do to the exponents?
Add them. Example: 10³ × 10⁴ = 10⁷.
When you divide powers of ten, what do you do to the exponents?
Subtract them. Example: 10⁸ ÷ 10³ = 10⁵.
To raise a power of ten to a power, what do you do?
Multiply the exponents. Example: (10⁴)² = 10⁸.
How do you multiply two numbers in standard form?
Multiply the coefficients, add the powers of ten, then re-normalise. Example: (3×10⁴)(2×10³) = 6×10⁷.
Find (2 × 10³)² without a calculator.
4 × 10⁶ — square the coefficient (2² = 4) and double the exponent.
A cube has edge 3 × 10² cm. Find its volume in standard form.
(3×10²)³ = 27×10⁶ = 2.7 × 10⁷ cm³.
After multiplying you get 0.5 × 10⁻³. Fix it.
5 × 10⁻⁴ — 0.5 = 5 × 10⁻¹, so subtract 1 from the exponent.
After cubing you get 27 × 10⁶. Fix it.
2.7 × 10⁷ — 27 = 2.7 × 10¹, so add 1 to the exponent.
What is an arithmetic sequence?
A sequence where each term differs from the previous one by a constant amount, the common difference d. Example: 4, 7, 10, 13 has d = 3.
What is the common difference, and how do you find it?
The constant gap between consecutive terms: d = uₙ − uₙ₋₁. Subtract any term from the next. Example: in 9, 5, 1 the difference is d = −4.
An arithmetic question asks for 'the value of u₂₀' in one part and 'which term equals 100' in another — how do you tell them apart?
Both use uₙ = u₁ + (n − 1)d. Want a VALUE (u₂₀)? Put n = 20 and compute. Want a POSITION ('which term = 100')? Set the formula equal to that value and solve for n. Spot value-vs-position first.
Why is it (n − 1)d and not nd in the nth-term formula?
You start at u₁ and add d only on each step after the first, so reaching the nth term takes (n − 1) steps. Example: u₅ = u₁ + 4d.
How do you find d from two terms, e.g. u₃ = 17 and u₇ = 41?
Divide the difference by the number of steps between them: (41 − 17) ÷ (7 − 3) = 24 ÷ 4 = 6.
How do you find which term equals a given value?
Set uₙ = u₁ + (n − 1)d equal to the value and solve for n. Example: 4 + (n−1)5 = 99 ⇒ n = 20.
Given a rule like uₙ = 20 − 4n, how do you read u₁ and d?
Substitute n = 1 for the first term (u₁ = 16); the coefficient of n is the common difference (d = −4).
When are three terms u₁, u₂, u₃ arithmetic?
When the differences are equal: u₂ − u₁ = u₃ − u₂. The middle term is the average of its neighbours.
How do you find an unknown k so that k+2, 2k+3, 5k−2 are arithmetic?
Set u₂ − u₁ = u₃ − u₂ and solve: (k + 1) = (3k − 5) ⇒ k = 3.
What is the difference between a sequence and a series?
A sequence is the list of terms (3, 7, 11, …); a series is their sum (3 + 7 + 11 + …).
In a word problem, do you add d n times or (n − 1) times?
Spot u₁ first. If you want a TERM ('the 6th row'), it's n − 1 jumps (row 1 = 0 jumps). If u₁ is a starting amount and you want the value 'after n years/steps', it's n jumps. Always ask: how many times do I add d to get there?
You are given u₁ = 7 and d = 4 and asked for the sum of the first 20 terms. What do you reach for — and what is the time-trap?
Go straight to Sₙ = (n/2)(2u₁ + (n − 1)d): S₂₀ = 10(14 + 19×4) = 900. Trap: do not waste time finding u₂₀ first — the u₁-and-d form needs only what you are given.
A question asks 'how many terms until the running total first passes 500?'. How do you set it up?
It is a TOTAL, so use the sum: set Sₙ > 500 and solve for n, then round UP to the next whole number (on Paper 2, scan the GDC table of Sₙ). Spot 'total/altogether' ⇒ Sₙ, not uₙ.
How do you choose which sum formula to use?
Know u₁ and d → use (n/2)(2u₁ + (n − 1)d). Know u₁ and the last term → use (n/2)(u₁ + uₙ).
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