Add the same amount each step and you get an AP — the pattern behind salary raises, stacked logs, and saving a little more every month. Two formulas unlock all of it.
Play with it
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.
Learn
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.
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).
To reach the nth term you add d a total of (n − 1) times to a. That gives the formula:
Find the 10th term of 2, 5, 8, …
Which term of 3, 8, 13, … equals 78?
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:
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.
Why this matters
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.
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 & sumLogs 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 APA balance under simple interest increases by the same amount yearly — a textbook AP.
"Add 5 minutes each week" running plans are APs; the total distance is a sum of an AP.
Rows that increase by a fixed number of seats — total capacity is Sₙ.
Simple interest, training plans, stadium seating and more — each explained with a diagram. Free to unlock.
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Modelled on CBSE's competency pattern — MCQ, assertion–reason and case-study items.
Interactive visualiser hand-built for trykarkedekho. Content from the rationalised NCERT Class 10 Maths syllabus (ncert.nic.in).
Buffyyour study buddy Buffy is an AI helper and can be wrong — always check your NCERT textbook.