What is the name of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10?

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The sequence 1, 2, 3, 4, 5 represents the foundation of many mathematical concepts. This comprehensive guide covers all essential formulas related to consecutive integers, arithmetic sequences, and their applications.

Core Formulas Table

Formula Type	Mathematical Expression	Description	Example (1-5)	Result
Sum of First n Natural Numbers	Sₙ = n(n+1)/2	Sum of consecutive integers from 1 to n	S₅ = 5(6)/2	15
Sum of Squares	Sₙ² = n(n+1)(2n+1)/6	Sum of squares from 1² to n²	S₅² = 5(6)(11)/6	55
Sum of Cubes	Sₙ³ = [n(n+1)/2]²	Sum of cubes from 1³ to n³	S₅³ = [15]²	225
Arithmetic Sequence	aₙ = a₁ + (n-1)d	nth term where d=1	a₅ = 1 + (5-1)×1	5
Arithmetic Series	Sₙ = n/2[2a₁ + (n-1)d]	Sum of arithmetic sequence	S₅ = 5/2[2 + 4]	15
Average Formula	Average = (First + Last)/2	Mean of consecutive integers	(1 + 5)/2	3
Product Formula	P = n!	Product of first n natural numbers	5! = 5×4×3×2×1	120

For Sequence 1-2-3-4-5-6-7-8-9-10

Property	Formula	Calculation	Result
Sum	S₁₀ = 10(11)/2	10 × 11 ÷ 2	55
Sum of Squares	S₁₀² = 10(11)(21)/6	10 × 11 × 21 ÷ 6	385
Sum of Cubes	S₁₀³ = [55]²	55²	3025
Average	(1 + 10)/2	11 ÷ 2	5.5

For Sequence 1-2-3-4-5-6-7

Property	Formula	Calculation	Result
Sum	S₇ = 7(8)/2	7 × 8 ÷ 2	28
Sum of Squares	S₇² = 7(8)(15)/6	7 × 8 × 15 ÷ 6	140
Average	(1 + 7)/2	8 ÷ 2	4

Special Pattern Formulas

Triangular Numbers (1, 3, 6, 10, 15...)

Formula: Tₙ = n(n+1)/2
For position 5: T₅ = 5(6)/2 = 15

Square Numbers (1, 4, 9, 16, 25...)

Formula: Sₙ = n²
For position 5: S₅ = 5² = 25

Pentagonal Numbers

Formula: Pₙ = n(3n-1)/2
For position 5: P₅ = 5(14)/2 = 35

Advanced Applications

Sum from 1 to 100 Formula

Formula: S₁₀₀ = 100(101)/2 = 5,050
Historical Note: This is famously attributed to Gauss's childhood calculation

General Range Formula (Sum from a to b)

Formula: Sum = (b-a+1)(a+b)/2
Example (1 to 5): (5-1+1)(1+5)/2 = 5×6/2 = 15

Consecutive Even Numbers (2, 4, 6, 8, 10)

Formula: Sum = n(n+1)
For 5 terms: 5(6) = 30

Consecutive Odd Numbers (1, 3, 5, 7, 9)

Formula: Sum = n²
For 5 terms: 5² = 25

Mathematical Relationships

Key Properties of 1-2-3-4-5 Sequence

Symmetry: The sequence is symmetric around its median (3)
Linear Growth: Each term increases by 1
Sum Property: Sum equals the middle term × number of terms
Perfect Relationships:
- 1 + 5 = 2 + 4 (outer terms equal inner terms)
- Sum of cubes equals square of sum

Important Identities

Identity	Mathematical Expression	Verification
Sum = Middle × Count	For odd n: Sum = middle × n	3 × 5 = 15 ✓
Cube Sum Identity	(1³ + 2³ + ... + n³) = (1 + 2 + ... + n)²	225 = 15² ✓
Even-Odd Relationship	Sum of n odds = n²	1+3+5+7+9 = 25 = 5² ✓

What is the name of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10?

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