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Lesson 8 of 8

Predicting What Comes Next: Exploring Sequences and Progressions · Lesson 8 of 8

Chapter Summary and Practice

“Find the pattern, predict the term and make sure the sequence of answers adds up.”

Learning Objectives

• Recall the language of sequences. • Choose between explicit and recursive descriptions. • Identify and solve AP and GP problems. • Use the natural-number sum formula. • Connect sequence rules with visual and real-life patterns.

Sequences help us describe ordered patterns and predict later terms. This chapter moved from simple term notation to explicit and recursive rules, then focused on two important families: arithmetic and geometric progressions.

Sequences and Term Notation

A sequence is an ordered list. The notation tₙ refers to the term in position n. Sequences can be finite or infinite.

Explicit and Recursive Rules

An explicit rule gives a term directly from n. A recursive rule builds a term from previous terms and therefore requires starting information.

Arithmetic Progressions

AP nth termLaTeX

An AP has constant difference d. It may increase or decrease depending on the sign of d.

Sum of Natural Numbers

Natural-number sumLaTeX

The same formula gives the nth triangular number.

Geometric Progressions

GP nth termLaTeX

A GP has constant ratio r. Ratios greater than 1 can produce rapid growth; ratios between 0 and 1 produce decay.

How to Choose the Right Method

ClueUse
Term depends directly on positionExplicit rule
Term described from previous term(s)Recursive rule
Same amount added/subtractedAP
Same factor multipliedGP
1+2+…+nn(n+1)/2
Repeated percentage/fraction of previous amountGP
Common Mistakes

• Treating the subscript in tₙ as multiplication, • Forgetting that sequence position n must be a natural number, • Using AP formulas when the ratio—not the difference—is constant, • Using GP formulas when the difference is constant, • Forgetting the n−1 in both AP and GP nth-term formulas, • Writing a recursive rule without the required starting term, • Assuming every visible pattern has a unique simple rule,

Guided Practice

Guided Example 1: Explicit Rule

Problem
Find the 40th term of tₙ=3n−4.

  1. 1.Substitute n=40.
  2. 2.t₄₀=3(40)−4=116.
Guided Example 2: AP

Problem
Find the 18th term of 5,9,13,17,… .

  1. 1.a=5,d=4.
  2. 2.t₁₈=5+17×4=73.
Guided Example 3: Natural-Number Sum

Problem
Find 1+2+…+50.

  1. 1.S₅₀=50×51/2.
  2. 2.S₅₀=1275.
Guided Example 4: GP

Problem
Find the 7th term of 2,6,18,… .

  1. 1.a=2,r=3.
  2. 2.t₇=2×3⁶.
  3. 3.t₇=1458.
Guided Example 5: Recursive Sequence

Problem
If t₁=2 and tₙ=tₙ₋₁+5, find t₅.

  1. 1.t₂=7.
  2. 2.t₃=12.
  3. 3.t₄=17.
  4. 4.t₅=22.

Practice Problems

Practice Questions
  1. Find the first five terms of tₙ=4n−5.
  2. Find the 12th and 20th terms of tₙ=6n−3.
  3. Determine whether 203 is a term of the sequence tₙ=5n−2.
  4. Generate five terms from t₁=−3, tₙ₊₁=tₙ+4.
  5. Write a recursive rule for 8,13,18,23,… .
  6. Find the 25th term of the AP 4,10,16,22,… .
  7. Which term of the AP 20,16,12,8,… is −60?
  8. A worker earns ₹5,20,000 in Year 1 and receives a ₹30,000 annual increase. In which year will the salary first reach ₹7,00,000?
  9. Find the sum of the first 40 natural numbers.
  10. Find 22+23+…+60.
  11. A triangular arrangement has 35 rows. How many objects are used?
  12. Find the 9th term of 3,9,27,… .
  13. Find the nth term of 8,4,2,1,… .
  14. A bacteria culture starts with 25 bacteria and doubles every hour. Write a rule for the number after n stages.
  15. A bouncing ball reaches 70% of its previous height. If its first rebound is 14 m, find the height of the fifth rebound.

Quiz

Quick check

Which sequence is an AP?

Quick check

Which sequence is a GP?

Quick check

What does an explicit formula allow you to do?

Quick check

What is the sum 1+2+…+n?

Quick check

If a GP has 0<r<1, what usually happens to positive terms?

Before You Finish the Chapter

Check that you can read tₙ notation, distinguish finite and infinite sequences, use explicit and recursive rules, identify APs and GPs, calculate nth terms, use the sum n(n+1)/2, and explain why AP and GP graphs behave differently.

Key Takeaways

Key Takeaways

• Sequences are ordered patterns. • Explicit rules use position; recursive rules use earlier terms. • APs have constant difference and use tₙ=a+(n−1)d. • The first n natural numbers sum to n(n+1)/2. • GPs have constant ratio and use tₙ=arⁿ⁻¹. • Repeated multiplication explains rapid growth and geometric decay.

Coming Next

This completes the chapter. Continue by practising how to recognise whether a pattern is additive, multiplicative, recursive or position-based.

Previous · Lesson 7

Geometric Progressions in Fractals and Bounces

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