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Quadratic Equation Till N

Math

a

1

b

3

c

-4

Highlight

Presets

nth term

f(n) = n² + 3n − 4

n = 1 to 10

Sum till N (10 terms)

510

Direct total and closed form agree

Second difference

2

Constant, and equal to 2a

Average term

51

Min 0 at n = 1 · Max 126 at n = 10

Sequence

0, 6, 14, 24, 36, 50, 66, 84, 104, 126

Parabola and terms

f(n) curve

sequence terms

vertex and axis of symmetry

real roots

Difference staircase

f(n)

0
6
14
24
36
50
66
84
104

Δ₁

6
8
10
12
14
16
18
20

Δ₂

2
2
2
2
2
2
2

Every Δ₂ equals 2 = 2a = 2 × 1. Only the first rows are drawn.

Term table

Rows per page

nf(n)Δ₁Δ₂Running sum
10——0
266—6
3148220
42410244
53612280
650142130
766162196
884182280
9104202384
10126222510

Roots, turning point and shape

Discriminant Δ = b² − 4ac

25

Positive — the parabola crosses the n-axis twice

Roots of f(n) = 0

n = 1, n = -4

Vertex (turning point)

(-1.5, -6.25)

Axis of symmetry

n = -1.5

Direction

Opens upward (a > 0) — the sequence has a minimum

Behaviour across the range

Increasing at every step

Vertex form

(n + 1.5)² − 6.25

Factored form

(n − 1)(n + 4)

Closed-form sum check

510

Matches the running total exactly

f(0) (constant term)

-4

In vertex form the same sequence is f(n) = (n + 1.5)² − 6.25, so h = -1.5 and k = -6.25.

Jump to a term, or search for a value

Evaluates f(k) directly — no need to list every row.

f(250) = 63,246 (outside the generated range)

Solves f(n) = target, so a hit outside the table is still reported.

Found at n = 10 (row 10). It is also reached at n = -13, outside the generated range.

Copy the results out

f(n) = n^{2} + 3n - 4

About This Tool

Quadratic Equation Till N – Generate and Analyse Quadratic Sequences

A quadratic sequence is what you get when a quadratic expression is evaluated at consecutive whole numbers. Feed f(n) = n² + 3n − 4 the indices 1 to 10 and it produces 0, 6, 14, 24, 36, 50, 66, 84, 104, 126. This quadratic sequence calculator builds that list for any coefficients you like, then shows the differences, the running total and the shape of the parabola the terms are sampled from.

How the terms are generated

Every row is a direct substitution. For n = 3 the tool computes 1(9) + 3(3) − 4 = 14, and the step-by-step panel prints that working for the opening terms so it can be copied into a homework write-up. You control where the sequence starts, how far it runs, and how big each step is, so f(0), f(2), f(4), … is as easy to produce as the usual n = 1, 2, 3, ….

Why the second difference matters

Subtracting neighbouring terms gives the first differences, which for a quadratic work out as a(2n + 1) + b — still growing. Do it once more and the n cancels, leaving the second difference 2a, the same on every row. That constant is the fingerprint of a quadratic sequence, and the difference staircase in the tool lays the three rows out the way a textbook does so the pattern is impossible to miss.

Reading the staircase
A constant bottom row means quadratic. A constant middle row instead means the sequence is arithmetic — linear, not quadratic — which happens here exactly when a = 0.

Summing the series two ways

The table carries a cumulative column, but adding ten numbers by hand is not the point of the exercise. The closed-form identity Σ f(n) = a·N(N + 1)(2N + 1)/6 + b·N(N + 1)/2 + c·N reaches the same total in one line. For the example above that is 385 + 165 − 40 = 510, matching the running sum exactly. Both values are shown side by side, which turns the sum of a quadratic series into something you can verify rather than trust.

Working backwards to the nth term

Exam questions often run the other way: here are the terms, find the rule. Paste a list such as 0, 6, 16, 30, 48 into the terms-to-nth-term mode and the finite-difference method does the rest. The second difference is 4, so a = 2; the first gap fixes b; the opening term fixes c; and the result f(n) = 2n² − 2 is checked against every value you supplied. If the second differences are not constant, the tool says the sequence is not quadratic and prints the differences so you can see why.

Beyond the table

Because the same coefficients describe a parabola, the tool also reports the discriminant, the real or complex roots, the vertex, the axis of symmetry and whether the curve opens upward or downward. The chart draws the continuous curve with the sequence terms marked as dots on it, which is the clearest way to see that a sequence is just a sampled function. Single-term lookup answers "what is the 250th term?" without listing everything in between, and the value search solves f(n) = target algebraically, so it can tell you a number appears at n = 250 even when your table stops at 10.

Very large terms
Beyond roughly 9 × 10¹⁵ a term can no longer be stored exactly as a double, so the display switches to scientific notation and the final digits become approximate. Keep N and the start index modest when you need exact integers.

Coefficients accept decimals and fractions such as 3/4, the equation box parses textbook notation like 2n^2 - 5n + 3, and results export as a comma list, a plain list, a LaTeX array or a CSV file for a spreadsheet.

Frequently Asked Questions

Is the Quadratic Equation Till N free?

Yes, Quadratic Equation Till N is totally free :)

Can I use the Quadratic Equation Till N offline?

Yes, you can install the webapp as PWA.

Is it safe to use Quadratic Equation Till N?

Yes, any data related to Quadratic Equation Till N only stored in your browser (if storage required). You can simply clear browser cache to clear all the stored data. We do not store any data on server.

How does this quadratic sequence generator work?

You give it the coefficients a, b and c (or an equation, a vertex form, or a list of terms) and a term count N. It then evaluates f(n) = an² + bn + c at each index in turn, collecting the value, the first difference, the second difference and the running sum for every row. All of the arithmetic happens in your browser, so nothing you type is sent anywhere.

Why is the second difference always the same number?

Subtracting f(n) from f(n + 1) leaves a(2n + 1) + b, which still depends on n. Subtracting one of those from the next cancels the n entirely and leaves 2a. That is why a constant second difference identifies a quadratic sequence, and why the constant is exactly twice the coefficient of n². If you generate the sequence with a step larger than 1, the second difference becomes 2a × step².

How does the tool find the nth-term rule from a list of terms?

It uses the finite-difference method. The second difference gives a = Δ₂ ÷ 2, the first gap between terms then gives b, and the first term gives c. The derivation is shown step by step, and the recovered formula is checked against every term you supplied. If the second differences are not constant the tool says so and prints them, rather than returning a formula that does not fit.

Is the sum of the series calculated by adding the terms up?

Both ways, deliberately. The table carries a running total, and the closed-form identity Σ f(n) = a·N(N + 1)(2N + 1)/6 + b·N(N + 1)/2 + c·N is evaluated separately so you can check the shortcut against the long way. With whole-number coefficients the two agree exactly; with fractions they can differ in the last decimal place because of floating-point rounding.

Can I start at n = 0, use negative indices, or count in steps of 2?

Yes. The start index may be zero or negative, and the step can be any whole number from 1 to 1000, so sequences like f(0), f(2), f(4), … are generated as easily as f(1), f(2), f(3), …. The differences, the sum and the closed-form check all adapt to the range you choose.

What is the difference between this and a quadratic equation solver?

A solver answers one question — which values of x make ax² + bx + c equal zero. This tool treats the same expression as a sequence generator: it tabulates the terms, analyses their differences, sums them, and works backwards from terms to a formula. Roots, discriminant and vertex still appear here, but as a secondary readout beneath the term table.