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[HO] SUMS AND SEQUENCES

Sums and Sequences

Tags: #◇Problem Solving #◇Sequences #◇Summations #◇CCDSTRU #◇M4

◇Important Formulas & Concepts

Sequences

A sequence {an}\{a_n\} is an ordered list (a function from a subset of integers). Geometric Progression: Sequence of the form a,ar,ar2,…,arn,…a, ar, ar^2, \dots, ar^n, \dots
Initial term aa, common ratio rr. Arithmetic Progression: Sequence of the form a,a+d,a+2d,…,a+nd,…a, a+d, a+2d, \dots, a+nd, \dots
Initial term aa, common difference dd. Recurrence Relation: An equation that expresses ana_n in terms of one or more of the previous terms (e.g., an=3an−1+2a_n = 3a_{n-1} + 2).

Summations

The sum of terms am,am+1,…,ana_m, a_{m+1}, \dots, a_n is denoted by ∑j=mnaj\sum_{j=m}^n a_j.

Common Closed Form Summations: ∑j=mn1=n−m+1\sum_{j=m}^n 1 = n - m + 1 ∑j=0nrj=rn+1−1r−1(where r≠1)\sum_{j=0}^n r^j = \frac{r^{n+1} - 1}{r - 1} \quad (\text{where } r \neq 1) ∑j=0nj=n(n+1)2\sum_{j=0}^n j = \frac{n(n + 1)}{2} ∑j=0nj2=n(n+1)(2n+1)6\sum_{j=0}^n j^2 = \frac{n(n + 1)(2n + 1)}{6} ∑j=0nj3=n2(n+1)24\sum_{j=0}^n j^3 = \frac{n^2(n + 1)^2}{4}

◇Worked Examples

Example 1: Expanding Sums
∑j=150(2j)=2+4+6+⋯+100\sum_{j=1}^{50} (2j) = 2 + 4 + 6 + \dots + 100

Example 2: Splitting Summations
∑j=0n(j2+3j)=∑j=0nj2+∑j=0n3j\sum_{j=0}^{n} (j^2 + 3j) = \sum_{j=0}^{n} j^2 + \sum_{j=0}^{n} 3j

Example 3: Double Summation Evaluation
Evaluate ∑j=1n∑k=1njk\sum_{j=1}^n \sum_{k=1}^n jk Inner sum: ∑k=1njk=j∑k=1nk=j(n(n+1)2)\sum_{k=1}^n jk = j \sum_{k=1}^n k = j \left( \frac{n(n+1)}{2} \right) Outer sum: ∑j=1n(jn(n+1)2)=n(n+1)2∑j=1nj=n(n+1)2⋅n(n+1)2\sum_{j=1}^n \left( j \frac{n(n+1)}{2} \right) = \frac{n(n+1)}{2} \sum_{j=1}^n j = \frac{n(n+1)}{2} \cdot \frac{n(n+1)}{2} Final result: n2(n+1)24\frac{n^2(n+1)^2}{4}

◇References & Navigation