Number Theory and Sequences: Fibonacci, Combinations, and Basic Operations
The Fibonacci sequence, combinations with repetition, sums of consecutive integers, reciprocals, and absolute value — five foundational number theory calculations.
From ancient number sequences to basic arithmetic operations, these five calculations show up throughout math coursework and problem-solving.
Fibonacci sequence: each term built from the two before it
Starting from F(1) = F(2) = 1, each subsequent term is the sum of the previous two: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 — the 10th term is 55. Close approximations of this sequence appear in nature, from sunflower seed spirals to nautilus shells.
Combinations with repetition: counting when repeats are allowed
Choosing 3 scoops from 5 ice cream flavors, where repeating a flavor is allowed, gives C(5+3-1, 3) = C(7,3) = 35 possible combinations — more than the 10 you'd get if repeats weren't allowed, since repeated selections open up additional possibilities.
Sum of consecutive integers: a shortcut for adding a range
Rather than adding every number individually, the sum from 1 to 10 uses n × (first + last) / 2 = 10 × 11 / 2 = 55 — instant even for ranges spanning millions of terms.
Reciprocal and absolute value: two of the most basic operations
The reciprocal of 8 is 1/8 = 0.125 (undefined for zero); the absolute value of −15 is 15. Simple as they are, both operations underpin countless other formulas — reciprocals appear throughout rate and ratio calculations, and absolute value appears anywhere distance or magnitude matters regardless of direction.
Building blocks worth knowing well
Sequences, combinatorics, and basic operations form the foundation that more advanced math builds on. Try the Fibonacci sequence calculator, combinations with repetition calculator, sum of consecutive integers calculator, reciprocal calculator, and absolute value calculator.