Tutorial on sequences and summation notation with examples and practice problems.
A sequence is an ordered list of numbers, which can be finite or infinite. Terms are typically denoted with subscripts: \( a_1, a_2, a_3, \ldots, a_n, \ldots \) where \( a_n \) represents the nth term.
\( 2, 4, 6, 8, 10, \ldots \)
Here \( a_1 = 2, a_2 = 4, a_3 = 6 \), etc.
\( \frac{1}{3}, \frac{1}{5}, \frac{1}{7}, \frac{1}{9}, \frac{1}{11}, \ldots \)
The first 4 terms of \( a_n = 2n + (-1)^n \):
\( a_1 = 2(1) + (-1)^1 = 1 \)
\( a_2 = 2(2) + (-1)^2 = 5 \)
\( a_3 = 2(3) + (-1)^3 = 5 \)
\( a_4 = 2(4) + (-1)^4 = 9 \)
Find the first 5 terms of:
\( a_1 = 1, \quad a_2 = 3, \quad a_n = 2a_{n-1} - a_{n-2} \text{ for } n > 2 \)
\( a_1 = 1 \) (given)
\( a_2 = 3 \) (given)
\( a_3 = 2a_2 - a_1 = 2(3) - 1 = 5 \)
\( a_4 = 2a_3 - a_2 = 2(5) - 3 = 7 \)
\( a_5 = 2a_4 - a_3 = 2(7) - 5 = 9 \)
The sum of terms \( a_1, a_2, \ldots, a_n \) is written using sigma notation:
\[ \sum_{i=1}^{n} a_i = a_1 + a_2 + a_3 + \cdots + a_n \]
Sum of the first 7 terms of \( 2, 4, 6, 8, 10, 12, 14, 16, \ldots \):
\[ S = 2 + 4 + 6 + 8 + 10 + 12 + 14 = 56 \]
Sum of first 5 terms where \( a_n = \frac{2n+1}{n} \):
\[ S = \sum_{i=1}^{5} \frac{2n+1}{n} = \frac{2\cdot1+1}{1} + \frac{2\cdot2+1}{2} + \frac{2\cdot3+1}{3} + \frac{2\cdot4+1}{4} + \frac{2\cdot5+1}{5} = \frac{737}{60} \]
Find the first 4 terms of \( a_n = n^2 - n \).
Find the first 6 terms of the recursive sequence:
\( a_1 = -1, \quad a_2 = -3, \quad a_n = -a_{n-1} - 2a_{n-2} \text{ for } n > 2 \)
Find the sum of the first 5 terms where \( a_n = n^2 + \frac{n+1}{2} \).
Exercise 1: \( 0, 2, 6, 12 \)
Exercise 2: \( -1, -3, 4, 0, -8, 8 \)
Exercise 3: \( 65 \)
More Math Problems with Solutions.