AP & Coordinate Geometry
Two whole Class 10 chapters in one tool. Build an arithmetic progression from a, d and n — or from a list of terms — and get the nth term, both sum formulas, missing terms and arithmetic means. Then drag points on a live grid for distance, section, slope, area and centroid, every answer with the substitution shown.
The nth term of an AP is aₙ = a + (n − 1)d, where a is the first term and d the common difference. For 5, 8, 11, …, the 20th term is 5 + 19 × 3 = 62. The sum of n terms is Sₙ = n/2 [2a + (n − 1)d], giving S₂₀ = 670.
Your AP
Fractions like 3/2 and decimals are allowed everywhere.
The sequence
First term a, common difference d.
nth term
an = a + (n − 1)d
Sum of n terms
Both formulas, same answer.
Term finder
Which term equals a given number, and how many terms there are when the last one is known.
Is it a term?
Number of terms
Arithmetic mean & inserting means
Put k equally spaced numbers between two values so the whole list is an AP.
Your GP
Class 11A geometric progression multiplies by the same ratio r each time.
The sequence
First term a, common ratio r.
nth term
an = a·rn−1
Sum of n terms
Sn = a(rn − 1) ÷ (r − 1)
Sum to infinity
S∞ = a ÷ (1 − r), and it only exists when |r| < 1.
Your points
Type the coordinates, or drag the points on the grid.
First box is x, second is y.
On the grid
Drag A, B or C — every answer updates as you move.
Results
Working — the two points A and B
Distance, midpoint, section formulas, slope and the equation of the line.
Working — the triangle ABC
Area, collinearity, centroid and the type of triangle.
Every formula you need for AP and coordinate geometry
These two chapters carry a large share of the Class 10 board paper. Each formula below is the one the tool uses, with the substitution shown line by line.
Pick a tab
Progressions for AP and GP, Coordinate Geometry for points on the plane.
Enter or detect
Type a, d and n — or paste a list of terms and let the tool find a and d.
Drag the points
On the grid, drag A, B and C; the distance, area and centroid follow instantly.
Copy or print
Take the full working into your notebook, or save the graph as a PNG.
Arithmetic progression
A sequence where every step adds the same number d.
an = a + (n − 1)d
Sn = n2[2a + (n − 1)d]
When the last term l is known the sum is shorter: Sn = n2(a + l). Both give the same answer because l = a + (n − 1)d.
Finding n
Rearranging an = a + (n − 1)d gives
n = an − ad + 1
If n comes out as a whole number that is 1 or more, the value is a term of the AP. If n is a fraction — say 12.5 — the number lies between two terms and is not in the AP at all. The same formula counts the terms when the last term is known: how many two-digit multiples of 3 are there? a = 12, d = 3, l = 99, so n = (99 − 12)/3 + 1 = 30.
Arithmetic mean
The arithmetic mean of A and B is simply their average, and it is the single number that sits between them in an AP.
AM = A + B2
To insert k means between A and B the list has k + 2 terms, so the gap is d = B − Ak + 1. Example: three means between 2 and 14 give d = 12 ÷ 4 = 3, so the list is 2, 5, 8, 11, 14.
Geometric progression Class 11
Here every step multiplies by the same ratio r.
an = a·rn−1
Sn = a(rn − 1)r − 1, r ≠ 1
When |r| < 1 the terms shrink and the total settles down: S∞ = a1 − r. That is why 0.9 + 0.09 + 0.009 + … adds up to exactly 1.
Distance, midpoint and section
AB = √[(x₂ − x₁)² + (y₂ − y₁)²]
M = (x₁+x₂2, y₁+y₂2)
The point dividing AB internally in the ratio m : n is (mx₂+nx₁m+n, my₂+ny₁m+n). For an external division the plus signs become minus signs and m + n becomes m − n, which is why m = n has no external point — the lines are parallel and never meet.
Area, collinearity and centroid
Area = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|
If that area is 0 the three points lie on one straight line — that is the quickest collinearity test there is.
The centroid, where the three medians meet, is just the average of the vertices: G = (x₁+x₂+x₃3, y₁+y₂+y₃3). A triangle is right-angled when the two shorter squared sides add up to the longest squared side — the tool checks that exactly, so no rounding error can fool it.
AP & Coordinate Geometry — your questions
How do I find the nth term of an AP?
Use aₙ = a + (n − 1)d, where a is the first term and d is the common difference found by subtracting any term from the next one. For 7, 11, 15, … the common difference is 4, so the 15th term is 7 + 14 × 4 = 63. If the terms decrease, d is negative.
How do you check whether a number is a term of an AP?
Rearrange the nth term formula to n = (aₙ − a) ÷ d + 1 and see what n comes out as. If n is a whole number of 1 or more, the number is a term; if it is a fraction such as 12.5, the value lies between two terms and is not in the AP at all.
What is the distance formula in coordinate geometry?
The distance between two points is AB = √[(x₂ − x₁)² + (y₂ − y₁)²], which comes straight from Pythagoras' theorem. For A(1, 2) and B(4, 6), AB = √(3² + 4²) = √25 = 5 units. Order does not matter, because the differences are squared.
How do I use the section formula?
A point dividing AB internally in the ratio m : n is ((mx₂ + nx₁) ÷ (m + n), (my₂ + ny₁) ÷ (m + n)). Note that the m goes with the second point's coordinates. When m = n = 1 this becomes the midpoint formula, ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2).
How do you prove three points are collinear?
Work out the area of the triangle they form with ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|. If the area is 0, the three points lie on one straight line. This is quicker and safer than comparing slopes, which breaks down when a line is vertical.
How many terms are there between two given terms of an AP?
Use n = (last − first) ÷ d + 1 to count the terms, then subtract 2 for the ones in between. For example, the two-digit multiples of 3 run from 12 to 99 with d = 3, so n = (99 − 12) ÷ 3 + 1 = 30 terms in all.
What is the difference between an AP and a GP?
In an arithmetic progression each step adds the same number d, so aₙ = a + (n − 1)d. In a geometric progression each step multiplies by the same ratio r, so aₙ = a·rⁿ⁻¹. The sequence 2, 5, 8, 11 is an AP with d = 3; 2, 6, 18, 54 is a GP with r = 3.