What is the formula for the nth term of an arithmetic sequence?

The nth term of an arithmetic sequence is aₙ = f + (n − 1)d, where f is the first term, d is the common difference, and n is the position. The key is (n−1) not n — because you count the number of steps taken from the first term.

What is the sum formula for an arithmetic sequence?

The sum of the first n terms is Sₙ = n/2 × [2f + (n−1)d]. If you already know the last term, the shortcut is Sₙ = n/2 × (first term + last term). Both formulas give the same answer.

How do I check if a number is a term of an arithmetic sequence?

Set aₙ equal to the number and solve for n using aₙ = f + (n−1)d. If n is a positive whole number, the number is in the sequence. If n is a fraction or negative, it is not a term.

What is the common difference in an arithmetic sequence?

The common difference (d) is the fixed amount added (or subtracted) from one term to get the next. Calculate it as: d = any term − the term before it. d can be negative if the sequence is decreasing.

Is arithmetic sequences important for Kerala SSLC exam?

Yes — Chapter 1 (Arithmetic Sequences) is one of the highest-scoring chapters in the Kerala SSLC Maths exam. Questions typically test the nth term formula, sum formula, finding f and d from given terms, and checking membership. Mastering these four question types covers all marks from this chapter.

Chapter 1 · Kerala SSLC Class 10 Maths

Arithmetic Sequences

Two ways to learn — a crisp formula reference, or a story that makes the maths feel obvious.

nth Term formulaSum formulaTesting membershipFinding f and d

Also useful:Practice Problems — Ch 1 (8 solved questions)1-on-1 Kerala SSLC Maths Tuition

Kerala SSLCClass 10MathematicsChapter 1⏱ 8 min crisp · 12 min story

Follow the story first. The maths will feel obvious by the end.

How this works

Each scene introduces one idea through a story you can picture. At the end of each scene, tap "The Maths" to see the formula behind what just happened. Reading the story first — before the formula — is the whole trick.

The Onam Bet

It's the last school day before Onam break. Arjun is packing his bag — already mentally on the cricket field — when his older sister Diya leans against the doorframe with a particular smile. The kind that usually means someone is about to lose money.

Diya

I have an Onam deal for you. Option A: I'll give you pocket money for all ten days. ₹10 on Day 1, ₹20 on Day 2, ₹30 on Day 3 — ten more rupees each day. Or Option B — ₹400 cash, right now, and we're done.

Arjun's brain starts working fast. ₹10, ₹20, ₹30... The last day is only ₹100. Total must be less than ₹500. ₹400 in hand is better.

Arjun

Option B. Cash.

Diya hands him four ₹100 notes without flinching. That's the warning sign Arjun misses.

That evening, Arjun calls his friend Rahul to brag. There's a long silence on the other end.

Rahul

Bro. Write out all ten amounts first.

Diya's offer — day by day

10→20→30→40→50→60→70→80→90→100…

Rahul

Now add them.

Arjun starts. 10+20 = 30. +30 = 60. +40 = 100. +50 = 150. +60 = 210. +70 = 280. +80 = 360. +90 = 450. +100 = 550.

He stares at the paper for a very long time.

₹550.

He had taken ₹400 and walked away from ₹550. He left ₹150 on the table because the numbers looked small at the start.

Rahul

Welcome to arithmetic sequences.

The Cricket Drill

Two months later, Arjun's cricket coach announces a fitness drill.

Coach

Day 1: 6 squats. Day 2: 9 squats. Day 3: 12 squats. Three more each day. Write your Day 30 target before you leave.

His teammates start writing out every term. Day 4: 15. Day 5: 18. Day 6: 21 …

Arjun counts the terms he'd have to write. Thirty of them. There has to be a shortcut.

He thinks about it like a staircase. Day 1 is the ground floor. Every day is one step of +3. To reach Day 30, he climbs 29 steps — not 30, because he's already standing on step 1.

Day 30 = Day 1 + (29 steps × 3 squats)

= 6 + 87

= 93 squats

Arjun writes "93" and puts his pen down while his teammates are still on Day 12.

The "staircase" mental image: You start on step 1. To get to step n, you climb (n−1) stairs, each of height d. Your height = where you started + (stairs climbed × step height).

Diya's Revenge

Diya has been watching Arjun use the formula and decides it's time for a rematch.

Diya

New challenge. I'm saving money. Month 1: ₹200. Month 2: ₹250. Month 3: ₹300. Fifty more each month. How much will I have saved in total after 12 months?

Okay, thinks Arjun. I need to add 12 terms. I could add them one by one, but that's slow. Last time I got tricked because I underestimated the total. Let me find the last term first.

Last term (Month 12):

a₁₂ = 200 + (12−1) × 50

= 200 + 550 = ₹750

Now he has to add: 200 + 250 + 300 + … + 750. Twelve numbers. He writes the sequence forward, then writes it again backwards, and stacks them:

Forward: 200, 250, 300, 350, 400, 450, 500, 550, 600, 650, 700, 750

Backward: 750, 700, 650, 600, 550, 500, 450, 400, 350, 300, 250, 200

Each pair adds to: 950 + 950 + 950 + … (12 pairs)

Every pair sums to 950. There are 12 terms, so 12 pairs of (forward, backward). But each term appears twice — so the real total is half of (12 × 950).

Total = 12/2 × (200 + 750)

= 6 × 950

= ₹5,700

Arjun says "₹5,700" before Diya expects it.

Diya

… Fine. Not bad.

This pairing trick is ancient. When the mathematician Gauss was 8 years old, his teacher told the class to add all numbers from 1 to 100 to keep them busy. Gauss handed in the answer in seconds — he'd used exactly this pairing method. 1+100 = 101, 2+99 = 101 … 50 pairs × 101 = 5050. Two thousand years of mathematicians, same trick.

The Detective Test

A week later, Diya tries one more trick.

Diya

My savings sequence is 200, 250, 300, 350 … Will there ever be a month where I save exactly ₹475?

Arjun thinks. Is ₹475 somewhere in that list? He doesn't write out every term — that could take forever. Instead, he assumes it's there and works backwards.

If ₹475 is the nth term, then:

475 = 200 + (n − 1) × 50

275 = (n − 1) × 50

n − 1 = 5.5

n = 6.5

n = 6.5. That's not a whole number. There's no "6.5th month". So ₹475 is not in the sequence — it falls between the 6th term (₹450) and the 7th term (₹500).

Arjun

No. There's no month where you save exactly ₹475. You'll jump from ₹450 to ₹500.

Diya

I hate that you're good at this now.

The Big Picture

Arithmetic sequences are everywhere once you start looking — savings plans, construction patterns, sports drills, festival schedules, salary increments, staircase designs. The maths has exactly four questions you'll ever be asked, and one formula family answers all of them:

What is the nth term?

aₙ = f + (n−1)d

What is the sum of n terms?

Sₙ = n/2 × [2f + (n−1)d]

Is x a term of this sequence?

Set aₙ = x, solve for n. Whole? Yes. Fraction? No.

Find f or d given two terms

Write two equations, subtract to eliminate one unknown

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