Sequences (OCR GCSE Maths): Revision Notes

Things You Need to Be Able to Do with Sequences:

Example Sequences: Let's look at some common types of sequences:

1. Linear Sequence (Addition/Subtraction) Sequence: $7,10,13,16,19,…$

Rule: The numbers are increasing by $3$ each time.

• This is an arithmetic sequence where each term is found by adding a fixed number to the previous term.

Predicting Next Numbers:

• Add $3$ to the last number:
• $19+3=22$
• $22+3=25$
• Next two numbers: $22,25$

2. Geometric Sequence (Multiplication/Division) Sequence: $3,6,12,24,48,…$

Rule: The numbers are doubling each time.

• This is a geometric sequence where each term is found by multiplying the previous term by a fixed number.

Predicting Next Numbers:

• Multiply the last number by $2$:
• $48×2=96$
• $96×2=192$
• Next two numbers: $96,192$

3. Complex Arithmetic Sequence (Varying Differences) Sequence: $200,190,181,173,166,…$

Rule: The numbers are decreasing with a pattern. The differences between the terms are decreasing by $1$ each time.

• Subtract $10$, then $9$, then $8$, and so on. Predicting Next Numbers:

• Subtract the next difference ($7$, then $6$):

• $166−7=160$

• $160−6=154$

• Next two numbers: $160,154$

4. Fibonacci Sequence Sequence: $1,1,2,3,5,8,13,…$

Rule: Each number is the sum of the two previous numbers.

• This sequence is known as the Fibonacci sequence.

Predicting Next Numbers:

• Add the last two numbers together:
• $8+13=21$
• $13+21=34$
• Next two numbers: $21,34$

Question: The sequence $5,11,23,47,…$follows a pattern. Describe the rule and find the next two terms.

Step-by-Step Solution:

5. Identify the Pattern:

• Look at the differences between terms:
• $11−5=6$
• $23−11=12$
• $47−23=24$
• The differences are doubling each time ($6, 12, 24$).

6. Describe the Rule:

• The difference between the terms doubles each time.

7. Predict the Next Numbers:

• The next difference should be $24×2=48$.
• Add $48$ to the last number:
• $47+48=95$
• The next difference should be $48×2=96$
• Add $96$ to $95$:
• $95+96=191$
• Next two numbers: $95,191$

Final Answer: The next two terms are $95$ and $191$.

Example 1: Find the $nth$ term of the sequence $13, 19, 25, 31, 37, ...$

8. Identify the Common Difference:

• The difference between the terms is $+6$. So, $a=6$.

9. Write the $6$ Times Table:

• List the first few multiples of $6$:

10. Compare to the Given Sequence:

• The given sequence is:

• Notice that each term in the sequence is $7$ more than the corresponding term in the $6$ times table.

11. Write the $nth$ Term Formula:

• The $nth$ term is $6n+7$.

12. Test the Formula:

• To find the $5th$ term $(n=5)$:

• The formula works, so the $nth$ term is $6n+7$.

Final Answer: The $nth$ term of the sequence is $6n+7$.

Example 2: Find the $nth$ term of the sequence $−20,−16,−12,−8,−4,…$

Step 1: Identify the Common Difference

• Look at the sequence and see how much you add or subtract to move from one term to the next.
• Here, you add $4$ each time.
• $−16−(−20)=4$
• $−12−(−16)=4$
• $−8−(−12)=4$
• So, the common difference $d$ is $+4$.

Step 2: Write the Multiples of the Common Difference

• Since the common difference is $4$, write down the $4$ times table under the sequence to help identify the $nth$ term:

Step 3: Determine the Constant Term ($b$)

• Compare the given sequence with the $4$ times table to see what you need to subtract or add to get from the multiples of $4$ to the actual sequence.
• To go from the $4$ times table to the actual sequence, you subtract $24$ from each term.
• $4−24=−20$
• $8−24=−16$
• $12−24=−12$
• $16−24=−8$
• $20−24=−4$
• So, the constant term $b$ is $−24$.

Step 4: Write the nth Term Formula

• Now that you have the common difference and the constant term, you can write the $nth$ term formula:

Step 5: Test the Formula

• Use the formula to find a specific term in the sequence to ensure it's correct.
• For example, to find the $5th$ term $(n=5):$

• The $5th$ term is $−4$, which matches the sequence.

Final Answer: The $nth$ term of the sequence is $4n−24$.

Example: Find the $100th$ term of the sequence. 13. Substitute $n=100$ into the formula:

14. Final Answer: The $100th$ term of the sequence is $376$.

Example 3: Find the $nth$ term of the Sequence: $21,16,11,6,1,…$

Step 1: Identify the Common Difference

• Observe how much you subtract each time to move from one term to the next.
• Here, you subtract $5$ each time:
• $16−21=−5$
• $11−16=−5$
• $6−11=−5$
• $1−6=−5$
• So, the common difference $d$ is $−5$.

Step 2: Write the Multiples of the Common Difference

• Since the common difference is $−5$, write down the multiples of $−5$ under the sequence to help identify the $nth$ term:

Step 3: Determine the Constant Term ($b$)

• Compare the given sequence with the multiples of $−5$ to see what you need to add or subtract to get from the multiples of $−5$ to the actual sequence.
• To go from the −$$5 times table to the actual sequence, you add $26$ to each term.
• $−5+26=21$
• $−10+26=16$
• $−15+26=11$
• $−20+26=6$
• $−25+26=1$
• So, the constant term $b$is $+26$.

Step 4: Write the $nth$ Term Formula

• Now that you have the common difference and the constant term, you can write the $nth$ term formula:

Step 5: Test the Formula

• Use the formula to find a specific term in the sequence to ensure it's correct.
• For example, to find the $6th$ term ($n=6$):

• The $6th$ term is$−4$, which would be the next term in the sequence if it continued.

Final Answer: The $nth$ term of the sequence is $−5n+26$

Example 1: Writing Out the First $5$ Terms Given $nth$ term rule:

Step-by-Step Solution:

15. $1st$ Term $(n = 1)$:

16. $2nd$ Term $(n = 2)$:

17. $3rd$ Term $(n = 3)$:

4. $4th$ Term $(n = 4)$:

5. $5th$ Term $(n = 5)$:

Result: The first $5$ terms of the sequence are $4,11,18,25,32$

Note: The difference between each term is $7$, which matches the coefficient of $n$ in the $nth$ term formula.

Example 2: Writing Out the First $5$ Terms Given $nth$ term rule:

Step-by-Step Solution:

18. $1st$ Term $(n = 1)$:

19. $2nd$ Term $(n = 2)$:

20. $3rd$ Term $(n = 3)$:

21. $4th$ Term $(n = 4)$:

22. $5th$ Term $(n = 5)$:

Result: The first $5$ terms of the sequence are $11,14,19,26,35$

Example Sequence:$−2,1,6,13,22,…$

Step 1: Write Out the Square Numbers

• List the first few square numbers ($n^2$) underneath the sequence:

Step 2: Determine What to Add or Subtract

• Compare each term in the sequence with the corresponding square number:
• $−2=1−3$
• $1=4−3$
• $6=9−3$
• $13=16−3$
• $22=25−3$
• You consistently subtract $3$ from the square number to get the sequence term.

Final $n$th Term Formula

• Since you subtract $3$ from each square number:

Final Answer: The $nth$ term of the sequence is $n2−3$