Key Concepts:
Sequence Definitions:
Sequence:
An ordered list of numbers arranged according to a specific rule or pattern. Each number in the sequence is called a term, and the position of a term is denoted by \(n\). Sequences can be defined by a general formula (explicit rule) or a recursive formula. For example: 5, 10, 15, 20,... is a sequence with first term \(a_1 = 5\), second term \(a_2 = 10\) and so on.General Formula:
A formula that defines the \(n\)th term of a sequence, \(a_n\), in terms of \(n\). For examplme \(a_n=n^2-2\) will give a sequence -1, 2, 7, 14, 23,...Recursive Formula:
A formula that defines each term of a sequence in relation to one or more previous terms. For example, \(a_1 = 2\), \(a_{n+1} = a_n + 3\) will give a sequence 2, 5, 8, 11,...Sequence Evaluation:
To evaluate a sequence means to find specific terms using the general formula (nth term) or recursive rule.Example:
Find the 5th term of the sequence defined by \(T_n = 3n - 2\): \[ T_5 = 3(5) - 2 = 15 - 2 = 13 \]Arithmetic Progression (AP):
A sequence where each term increases or decreases by a constant value, called the common difference \(d\). The general form is: \[ a, a+d, a+2d, \dots \]Nth Term of AP:
\[ a_n = a + (n-1)d \] where \(a\) is the first term, \(d\) is the common difference, and \(n\) is the term number.Sum of the First \(n\) Terms of AP:
\[ S_n = \frac{n}{2} [2a + (n-1)d] \]Geometric Progression (GP):
A sequence where each term is multiplied by a constant value, called the common ratio \(r\). The general form is: \[ a, ar, ar^2, \dots \]Nth Term of GP:
\[ a_n = ar^{n-1} \] where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.Sum of the First \(n\) Terms of GP:
\[ S_n = a \frac{1-r^n}{1-r}, \quad r \neq 1 \]Sum to Infinity of GP:
\[ S_\infty = \frac{a}{1-r}, \quad |r| < 1 \]Sum of Sequences:
The sum of the terms in a sequence, calculated using the formulas above depending on whether the sequence is arithmetic or geometric.Example 1: Evaluating Terms in a Sequence
If \( U_n = n(n^2 + 1) \), evaluate \( U_5 - U_4 \).Solution:
Step 1: Find \( U_5 \): \[ U_5 = 5(5^2 + 1) = 5(25 + 1) = 5 \times 26 = 130 \] Step 2: Find \( U_4 \): \[ U_4 = 4(4^2 + 1) = 4(16 + 1) = 4 \times 17 = 68 \] Step 3: Subtract: \[ U_5 - U_4 = 130 - 68 = \boxed{62} \]Example 2: Find the 10th term of the AP \(3, 7, 11, \dots\).
Solution:
Identify the values: \[ a = 3, \quad d = 7 - 3 = 4, \quad n = 10. \] Use the formula for the \(n\)th term: \[ a_n = a + (n-1)d = 3 + (10-1) \cdot 4 = 3 + 36 = 39. \] Therefore, the 10th term is 39.Example 3: Find the sum of the first 8 terms of the AP \(5, 9, 13, \dots\).
Solution:
Identify the values: \[ a = 5, \quad d = 9 - 5 = 4, \quad n = 8. \] Use the formula for the sum of the first \(n\) terms: \[ S_n = \frac{n}{2} [2a + (n-1)d] = \frac{8}{2} [2 \cdot 5 + (8-1) \cdot 4]. \] Simplify: \[ S_n = 4 [10 + 28] = 4 \cdot 38 = 152. \] Therefore, the sum of the first 8 terms is 152.Example 4: Find the 6th term of the GP \(2, 4, 8, \dots\).
Solution:
Identify the values: \[ a = 2, \quad r = 4 \div 2 = 2, \quad n = 6. \] Use the formula for the \(n\)th term: \[ a_n = ar^{n-1} = 2 \cdot 2^{6-1} = 2 \cdot 2^5 = 2 \cdot 32 = 64. \] Therefore, the 6th term is 64.Example 5: Find the sum to infinity of the GP \(3, 1.5, 0.75, \dots\).
Solution:
Identify the values: \[ a = 3, \quad r = 1.5 \div 3 = 0.5. \] Use the formula for the sum to infinity: \[ S_\infty = \frac{a}{1-r} = \frac{3}{1-0.5} = \frac{3}{0.5} = 6. \] Therefore, the sum to infinity is 6.