Count the steps from the first term
The nth term is aₙ = a + (n − 1)d.
Reena's salary again
Reena's salary is ₹8000 in year 1 and goes up by ₹500 every year.
Year 2: 8000 + 500. Year 3: 8000 + 500 + 500 = 8000 + 2 × 500. Year 4: 8000 + 3 × 500.
How many ₹500 raises have been added by the 15th year?
What this lesson covers
The idea
The nth term (general term) of an AP with first term a and common difference d is aₙ = a + (n – 1)d; in an AP of m terms, aₘ is the last term, also denoted l.
Reena's salary again
Reena's salary is ₹8000 in year 1 and goes up by ₹500 every year.
Year 2: 8000 + 500. Year 3: 8000 + 500 + 500 = 8000 + 2 × 500. Year 4: 8000 + 3 × 500.
How many ₹500 raises have been added by the 15th year?
- 15
- 14
- 16
How many steps to reach a term?
The first block is a. Each d block is one step. Move n and look at how many d blocks there are. Find the term of this AP that equals the target.
The nth term
aₙ = a + (n − 1)d is the nth term (or general term) of an AP with first term a and common difference d.
To go from the 1st term to the nth term you take n − 1 steps of d: 10 terms means 9 steps. For the AP 2, 7, 12, …: a₁₀ = 2 + 9 × 5 = 47.
If an AP has m terms, aₘ is its last term. It is sometimes written *l* (a small letter L).
Notes
aₙ = a + (n − 1)d: the nth term is the first term plus n − 1 steps of d. The last term of an AP with m terms is aₘ, also written *l*.
Check yourself
An AP has 12 terms. Which one is its last term, *l*?
Reena earns ₹8000 in year 1, with a raise of ₹500 each year. What is her monthly salary (in ₹) in the 25th year?
Answer: 20000
a₂₅ = 8000 + (25 − 1) × 500 = 8000 + 12000 = 20000.
Which term of the AP 21, 18, 15, … is −81?
Answer: 35
−81 = 21 − 3(n − 1), so −102 = −3(n − 1), so n − 1 = 34 and n = 35.
Is 301 a term of the list 5, 11, 17, 23, …?
- a₁₂ — correct. Yes! With 12 terms, the last one is the 12th term, a₁₂. It is also written *l*.
- a₁₁. a₁₁ is the second-last term. The last of 12 terms is a₁₂.
- a. a is the first term.
- No. 301 = 5 + (n − 1) × 6 gives n = 151/3, which is not a whole number. — correct. Yes! The position n must be a whole number, and 151/3 is not.
- Yes. It is the 50th term.. Check: a₅₀ = 5 + 49 × 6 = 299, not 301.
- Yes. The list grows, and 301 is bigger than 5.. Being bigger is not enough. The list has to hit 301 exactly, at a whole-number position.