How do you find a term of an arithmetic sequence without knowing the first term?

Use the Proportion Rule: the change in terms is always proportional to the change in position. If you know any two terms at any two positions, find the rate (term change ÷ position change), then scale it to reach any other position — you never need the first term specifically.

How do you find the sum of terms in an arithmetic sequence?

Two shortcuts: for an odd number of consecutive terms, the sum equals the middle term multiplied by the count of terms. For any number of terms where you know the first and last, pair the first with the last, the second with the second-last, and so on — every pair adds to the same total, so the sum is (pair total) × (number of pairs).

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

Subtract a known term from the number and check whether the result is a multiple of the common difference. If it is, the number is a term (its position can be worked out from how many steps of d it took). If not, the number simply isn't in the sequence.

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 proportion rule for finding terms, middle-term and position-pair sum tricks, finding f and d from given terms, and checking membership. Mastering these 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.

Proportion ruleMiddle-term sumsTesting 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. Tap "The Maths" at the end of each scene to see the formula. Read the story before the formula — that's the trick.

The Onam Bet

It's the last school day before Onam break. Arjun is packing his bag, already thinking about ten days of cricket, games, and doing absolutely nothing, 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 a 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.

₹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 and pins a sheet on the noticeboard — squat counts for all 30 days, one more each day than the day before.

Coach

Half the sheet got soaked in the rain overnight. All I can still read is Day 4: 15 squats, and Day 9: 35 squats. Figure out Day 30 before practice starts.

His teammates panic — without Day 1, how can they even start?

Arjun doesn't need Day 1 at all. From Day 4 to Day 9, the position moved 5 steps, and the squat count moved from 15 to 35 — a jump of 20. So each single step must be worth 20 ÷ 5 = 4 squats.

Position change (Day 4 → Day 9) = 5

Term change = 35 − 15 = 20

Rate = 20 ÷ 5 = 4 per day

Day 30 is 21 steps after Day 9 (30 − 9 = 21). So the count rises by 21 × 4 = 84 more squats from Day 9's 35.

Day 30 is 21 steps after Day 9 (30 − 9 = 21)

Day 30 = 35 + (21 × 4)

= 35 + 84 = 119 squats

Arjun writes "119" and puts his pen down while his teammates are still hunting for Day 1.

Diya's Revenge

Diya has been watching Arjun get faster at this 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 total after 12 months?

Arjun: I need to add 12 terms. Find the last term first.

Last term (Month 12):

Position change (1 → 12) = 11, term change = 11 × 50 = 550

Month 12 = 200 + 550 = ₹750

Forward: 200, 250, 300 … 700, 750

Backward: 750, 700, 650 … 250, 200

Every pair adds to: 950 (12 pairs total)

Every pair sums to ₹950. 12 terms, 6 pairs. Total:

Total = 950 × 6 pairs

= ₹5,700

Arjun

₹5,700

Diya

… Fine. Not bad.

This pairing trick is ancient. When Gauss was 8, his teacher told the class to add 1 to 100. Gauss answered in seconds: 1+100=101, 2+99=101 … 50 pairs × 101 = 5050.

Diya

Before the big total — quick one. Without adding anything, what's Month 3 plus Month 9?

Month 3 and Month 9 are each exactly 3 steps away from Month 6 — one before it, one after. Arjun doesn't need either value: two terms equally spaced around a known middle term always add up to twice that middle term.

Month 6 = ₹450 · Month 3 + Month 9 = 2 × 450 = ₹900

Diya

Fine — try Month 5 plus Month 7 then.

Same idea — one step before Month 6, one step after. Also ₹900. Diya notices the pattern before Arjun even says it.

Diya

…Every pair around Month 6 gives ₹900, doesn't it.

Diya

Fine, quicker one: what's the total saved just from Month 4 to Month 8?

Five months, and Month 6 sits exactly in the middle. Arjun doesn't add five numbers — he just multiplies the middle term by how many terms there are.

Month 6 = ₹450 · Sum (5 terms) = 450 × 5 = ₹2,250

Arjun

₹2,250 — didn't even need the first or last month for that one.

Diya

One last one. I know Month 4 plus Month 8 comes to ₹900. Without telling you either number — what's Month 5 plus Month 7?

Arjun checks the positions, not the amounts: 4 + 8 = 12, and 5 + 7 = 12 too. Same position-sum.

Arjun

₹900. Has to be — same position-sum always means same term-sum.

Diya

…Show-off.

The Detective Test

A week later, Diya tries one more trick.

Diya

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

Is ₹475 in the list? Arjun doesn't write out every term — he checks divisibility instead.

475 − 200 = 275

275 ÷ 50 = 5.5

Not a whole number

475 − 200 = 275. Is 275 a multiple of the common difference, 50? 275 ÷ 50 = 5.5 — not a whole number. So ₹475 is not in the sequence. It falls between Month 6 (₹450) and Month 7 (₹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, sports drills, festival schedules, salary increments. No formula to memorise — just proportion and the middle-term tricks.

Find a term from two other terms

term change ∝ position change

Sum of terms equidistant from a middle term

before + after = 2 × middle

Sum of an odd number of consecutive terms

middle term × count

Is x a term of this sequence?

Is (x − known term) a multiple of d?

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