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Find nth term formulae using the DiNO method for linear sequences and second differences for quadratics. IGCSE worked examples for grades 5-9.
For a linear sequence (constant first difference), use the DiNO method: find the common Difference d, write dn, then calculate the zero term c = first term minus d. The nth term is dn + c. For a quadratic sequence (constant second difference), the nth term takes the form an squared + bn + c. Use the three memorised results to find a, b and c: 2a = second difference, 3a + b = first first-difference, a + b + c = first term.
Source: Edexcel IGCSE Mathematics (9-1) Specification 4MA1
A linear sequence (also called an arithmetic sequence) has a constant difference between consecutive terms. The nth term formula always takes the form dn + c, where d is the common difference and c is the zero term.
DiNO is a three-step mnemonic that makes finding the nth term of any linear sequence reliable and fast:
The nth term is then dn + c. For example, given the sequence 7, 11, 15, 19 the common difference is 4, so the formula starts as 4n. The zero term is 7 - 4 = 3, giving the nth term as 4n + 3.
The zero term c represents the value the sequence would have at position n = 0 if extended backwards. Because the difference is constant, every step forward adds d, so the value at position n is exactly c + dn, which we write as dn + c. This is identical to the equation of a straight line y = mx + c, which is why linear sequences produce straight-line graphs when plotted.
A quadratic sequence is one where the first differences are not constant, but the second differences (the differences of the differences) are constant. The nth term takes the form an² + bn + c.
To find a, b and c you need three equations. These come directly from comparing your sequence to the general quadratic formula:
| Result | Meaning |
|---|---|
| 2a = second difference | Halve the second difference to find a |
| 3a + b = first first-difference | Substitute a, solve for b |
| a + b + c = first term | Substitute a and b, solve for c |
Once you have all three values, write the nth term as an² + bn + c. Always verify by substituting n = 1, 2 and 3 to check the outputs match the original sequence.
Quadratic sequences appear naturally in computing. An algorithm with quadratic time complexity O(n²) produces a quadratic sequence of operations as the input size grows: 1, 4, 9, 16, 25 operations for inputs of size 1, 2, 3, 4, 5. Understanding quadratic growth helps explain why some programs slow down dramatically with larger data sets.
Set the nth term formula equal to the target number and solve for n. If n is a positive integer, the number is a term in the sequence. If n is not a whole number, the target does not appear in the sequence.
For linear sequences this means solving a simple linear equation. For quadratic sequences you will need to solve a quadratic equation, usually with the quadratic formula. The key check is whether the discriminant (b² - 4ac) produces a perfect square, because only then will n be a whole number.
Find the nth term of the sequence: 5, 8, 11, 14, ...
Step 1 (Difference): d = 8 - 5 = 3
Step 2 (N-variable): Write 3n
Step 3 (Zero term): c = 5 - 3 = 2
Step 4 (Write formula): nth term = 3n + 2
Check: n=1: 3(1)+2 = 5. n=2: 3(2)+2 = 8. n=3: 3(3)+2 = 11. All match.
Answer: 3n + 2
Find the nth term of the sequence: 5, 12, 23, 38, 57, ...
Step 1 (First differences): 7, 11, 15, 19
Step 2 (Second differences): 4, 4, 4 — constant, so quadratic
Step 3 (Find a): 2a = 4, so a = 2
Step 4 (Find b): 3a + b = 7 (first first-difference), so 6 + b = 7, b = 1
Step 5 (Find c): a + b + c = 5 (first term), so 2 + 1 + c = 5, c = 2
Check: n=1: 2+1+2 = 5. n=2: 8+2+2 = 12. n=3: 18+3+2 = 23. All match.
Answer: 2n² + n + 2
Is 150 a term in the sequence whose nth term is n² + 4n - 5?
Step 1 (Set equal to 150): n² + 4n - 5 = 150
Step 2 (Rearrange): n² + 4n - 155 = 0
Step 3 (Apply quadratic formula): n = (-4 +/- sqrt(16 + 620)) / 2 = (-4 +/- sqrt(636)) / 2
Step 4 (Evaluate discriminant): sqrt(636) = 25.22... which is not a whole number
Conclusion: Since n is not a positive integer, 150 is NOT a term in this sequence.
Answer: No, 150 is not a term in the sequence
Confusing position and value: n is the position number (1st, 2nd, 3rd...) and the nth term formula gives the value at that position. When the question says "find the 10th term", substitute n = 10 into the formula — do not look for 10 in the sequence.
Negative sign errors in quadratics: When second differences are negative, a is negative. Students often lose the sign during the three-result process. Write each equation out fully and solve carefully.
Not verifying the formula: The quickest way to lose marks is to write an nth term that does not actually produce the given sequence. Always substitute n = 1, 2 and 3 to check. This takes seconds and catches arithmetic errors.
Applying DiNO to a quadratic sequence: If the first differences are not constant, the sequence is not linear and DiNO does not apply. Always check first differences before choosing a method.
Check the difference pattern first: Before writing anything, calculate both first and second differences. Constant first differences mean linear (use DiNO). Constant second differences mean quadratic (use the three-result method).
Memorise the three results: For quadratic sequences, the three results (2a = second diff, 3a + b = first first-diff, a + b + c = first term) should be committed to memory. They appear on Higher papers almost every series.
Show your difference table: Examiners award method marks for clearly laid-out difference tables. Write the original sequence, first differences below, and second differences below that.
Sequence membership = solve and check integer: When asked "Is X a term?", set the nth term equal to X and solve. State clearly whether n is a positive integer or not — this is where the mark is awarded.
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