The answer is no. The nearest Fibonacci numbers are 144 (the 12th term) and 233 (the 13th). But how do you check without generating the entire sequence?
There is a theorem: a positive integer m is a Fibonacci number if and only if 5m^2 + 4 or 5m^2 - 4 is a perfect square. For 150, we get 5 x 150^2 + 4 = 112,504 and 5 x 150^2 - 4 = 112,496. Neither is a perfect square (sqrt of 112,504 is roughly 335.4, not an integer). So 150 fails the test. For 144, the check gives 5 x 144^2 + 4 = 103,684 = 322^2. Perfect square. Confirmed Fibonacci.
That single test replaces a brute-force search through the entire sequence. The new Fibonacci Sequence Calculator runs this check instantly, and it does two more things: compute any term F(n) up to n = 1000, and generate a table of the first N terms showing how the ratio of consecutive numbers converges to the golden ratio.
F(10) = 55. The ratio F(10)/F(9) = 1.6176... already close to phi = 1.6180339887. By F(40), the ratio matches phi to over 15 decimal places. The calculator uses BigInt arithmetic, so even F(1000) with its 209 digits is computed exactly.
35 ice cream combinations, or why repetition changes everything
Five flavors, three scoops. You can repeat a flavor. How many distinct selections exist?
Without repetition, you pick 3 different flavors from 5: C(5, 3) = 10 options. With repetition allowed, the formula changes to C(n+k-1, k). Plug in n = 5, k = 3: C(7, 3) = 35 options. Allowing repeats multiplied the count by 3.5.
The intuition comes from the stars and bars method. Represent your 3 scoops as stars (*) and use 4 bars (|) to separate them into 5 flavor bins. The arrangement **|| means 2 scoops of flavor 1, 1 of flavor 2, 0 of the rest. Count the ways to place 3 stars among 7 total positions: C(7, 3) = 35.
The Combinations with Repetition Calculator shows the result alongside the standard combination count, the difference, and the ratio. It also interprets the magnitude: 35 is a short list you could write out by hand, but C(69, 20) = 7.89 quadrillion is a needle-in-a-haystack scenario.
This formula appears more often than you might expect. Distributing 10 identical candies among 4 children? Same problem: n = 4, k = 10, answer = C(13, 10) = 286 ways. Counting terms in a polynomial of degree d in n variables? C(n+d-1, d). Any problem that asks how many ways to put identical items into distinct bins is a multiset problem solved by this formula.
The simplest rule in geometry: 180 degrees
Every triangle on a flat surface has interior angles summing to exactly 180 degrees. Know two, and the third is determined. The Triangle Angle Calculator does this in two ways: enter two angles to find the missing one, or enter all three to verify they form a valid triangle.
Angle alpha = 45 degrees, angle beta = 65 degrees. The calculator returns gamma = 70 degrees and classifies the triangle as acute (all angles below 90). Change beta to 90 and you get an invalid result: 45 + 90 = 135, leaving gamma = 45, which would actually be valid. But enter 100 + 90 and the calculator flags it: sum exceeds 180, no triangle exists.
The classification is automatic: acute (all under 90), right (one equals 90), or obtuse (one exceeds 90). Decimal precision goes to 0.01 degrees, which is more than enough for construction, surveying, or homework.
The math chapter is complete
These three calculators bring the math category to 40 tools. From percentage calculations to matrix operations, from prime factorization to Fibonacci sequences, every core topic in school and university mathematics now has a dedicated calculator on the English version of LiczGrupa.
Tools discussed in this article
- Fibonacci Sequence Calculator - compute F(n), list first N terms, check if a number is Fibonacci
- Combinations with Repetition Calculator - calculate C(n+k-1, k) with comparison to standard combinations
- Triangle Angle Calculator - find missing angles and classify triangles by type
More math tools
- Combinations & Permutations Calculator - C(n,r), P(n,r) and variations
- Probability Calculator - event probability as fraction, percent and odds
- Average Calculator - arithmetic, weighted, geometric and harmonic mean
- Pythagorean Theorem Calculator - find missing sides in right triangles
- Factorial Calculator - compute n! for any non-negative integer
- Arithmetic Sequence Calculator - nth term, sum and common difference
- Geometric Sequence Calculator - growth, decay and compound patterns
- Powers & Roots Calculator - any base to any exponent or root
- Matrix Calculator - add, multiply, determinant, inverse for 2x2 and 3x3
Check for a specific position
- Fibonacci number 10 - 10th element
- Fibonacci number 20 - larger values
- Fibonacci number 50 - astronomical