Stage 9 - Q1 - 2 marks
Start at 3; square, then add 4. Write the first three terms.
Hint
Brackets distinguish squaring before and after the addition/subtraction.
Stage 9 - Q2 - 2 marks
Start at 5; square, then subtract 6. Write the first three terms.
Hint
Brackets distinguish squaring before and after the addition/subtraction.
Stage 9 - Q3 - 3 marks
The fourth term is \(57\). Each step multiplies by 2, then adds 3. Find the first term.
Hint
Going from term 4 back to term 1 takes three inverse steps.
Stage 9 - Q4 - 3 marks
The fourth term is \(65\). Each step multiplies by 2, then adds 3. Find the first term.
Hint
Going from term 4 back to term 1 takes three inverse steps.
Stage 9 - Q5 - 4 marks
The third term is 437; the rule is square then subtract 4. All three terms are positive. Find the first term.
Hint
The positive-term condition selects a root at each stage.
Stage 9 - Q6 - 3 marks
A rule squares an integer and subtracts 2. Its output is 23. Find every possible input.
Hint
There is no positive-input restriction.
Stage 9 - Q7 - 3 marks
Choose the missing constant so that starting at 4 gives a constant sequence under “square, then subtract k”.
Hint
Set one application of the rule equal to its input.
Stage 9 - Q8 - 4 marks
The fourth term is \(\frac{91}{16}\). The rule divides by 2 then adds 3. Find the first term.
Hint
Reverse both the operations and their order.
Stage 9 - Q9 - 4 marks
A sequence rule is “subtract 1, then square”. The second term is 16. Find both possible first terms.
Hint
The square occurs after subtraction, so undo it first.
Stage 9 - Q10 - 5 marks
Challenge: The rule “square then subtract 20” can produce a constant integer sequence. Find every possible starting integer and explain why there are no others.
Hint
Look for two integers with product minus the constant and sum minus one.