Arithmetic Progressions | Maths Grade 10 CBSE

Play with it

Build an AP, see the sum

Drag the first term a and the common difference d. Each bar is a term; watch the sequence tilt and the running sum Sₙ change with the formulas.

aₙ = a + (n − 1)ddrag a and d

first term a 2

common difference d 2

Try:

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The three ideas in this chapter

An arithmetic progression (AP) is a list of numbers where each term is obtained by adding a fixed number to the previous one. That fixed number is the common difference d; the starting value is the first term a.

So an AP looks like: a, a + d, a + 2d, a + 3d, … For example, 3, 7, 11, 15, … has a = 3 and d = 4.

Test for an AP

A sequence is an AP only if the difference between every pair of consecutive terms is the same. 2, 4, 8, 16 is not an AP (differences 2, 4, 8 grow).

Watch out: d can be negative (a decreasing AP like 10, 8, 6, …) or zero (a constant sequence). Both are still valid APs.

To reach the nth term you add d a total of (n − 1) times to a. That gives the formula:

aₙ = a + (n − 1)d

Worked example · find the 10th term

Find the 10th term of 2, 5, 8, …

Here a = 2 and d = 5 − 2 = 3.
a₁₀ = a + (10 − 1)d = 2 + 9×3.
= 2 + 27 = 29.

Worked example · which term is 78?

Which term of 3, 8, 13, … equals 78?

a = 3, d = 5. Set aₙ = 78: 3 + (n − 1)×5 = 78.
5(n − 1) = 75 → n − 1 = 15 → n = 16.
So 78 is the 16th term.

Common mistake: using n instead of (n − 1). The first term needs zero jumps of d, so the count is always (n − 1).

There's a beautiful trick (the one young Gauss used): write the sum forwards and backwards and add — every pair sums to (first + last). That gives:

Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l)

where l is the last term. Use the first form when you know a and d; the second when you know the first and last terms.

Worked example · sum of 1 to 100

This is an AP with a = 1, l = 100, n = 100.
S₁₀₀ = n/2 (a + l) = 100/2 × (1 + 100).
= 50 × 101 = 5050.

Common mistake: mixing the two sum formulas. If you only know a and d, use n/2[2a + (n − 1)d]; don't guess the last term.

Why this matters

Where you'll actually use this

Anything that grows by a fixed step is an AP — your salary with a yearly raise, a savings plan, rows of seats, stacked pipes. The sum formula tells you the total instantly.

Salary & savings that grow each year

Start at ₹18,000 and add a fixed ₹1,500 raise every year — that's an AP, and your salary in any year is just a + (n−1)d. Save a fixed amount more each month and the same formula tells you the total saved after n months without adding them one by one.

nth term & sum

Stacking & seating

Logs in a woodpile, pipes on a truck, seats in an auditorium that gain a few per row — all follow an AP. To count the total you don't add row by row; you use Sₙ = n/2 (first + last). Event planners and warehouses do exactly this.

Sum of an AP

🏦 Simple-interest growth

A balance under simple interest increases by the same amount yearly — a textbook AP.

🏃 Training plans

"Add 5 minutes each week" running plans are APs; the total distance is a sum of an AP.

🎟️ Stadium seating

Rows that increase by a fixed number of seats — total capacity is Sₙ.

🔒 More real-world applications

Simple interest, training plans, stadium seating and more — each explained with a diagram. Free to unlock.

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Check yourself

Competency quiz

Modelled on CBSE's competency pattern — MCQ, assertion–reason and case-study items.

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