Solving Equations Graphically and Arithmetic Sequences

Solving Equations and Inequalities by Graphing

• Linear and absolute value equations are solved graphically by setting each side equal to $y$, graphing both equations, and finding their point(s) of intersection:
  • For $-3x + 20 = 0.2x + 4$, graph $y = -3x + 20$ and $y = 0.2x + 4$. The lines intersect at $(5, 5)$, giving the solution $x = 5$.
  • For $|1x - 4| = \frac{1}{2}x + 1$, graph $y = |1x - 4|$ and $y = \frac{1}{2}x + 1$. The graphs intersect at $(2, 2)$ and $(10, 6)$, giving the solutions $x = 2$ and $x = 10$.
• Linear inequality word problems compare relative positions or distances over time:
  • For a motorcycle $40\,\text{mi}$ ahead traveling at $40\,\text{mph}$ and a car traveling at $60\,\text{mph}$, write the inequality $60x > 40x + 40$.
  • Solving yields $x > 2$, meaning the car will be ahead after $2\,\text{hours}$ (meeting at $(2, 120)$).

Arithmetic Sequences

• An arithmetic sequence is a sequence of numbers with a constant common difference $d$ (via addition or subtraction) between consecutive terms.
• Explicit Formula:
  • Formula: $a_n = a_1 + d(n - 1)$
  • Allows finding any term in a sequence without knowing the previous term.
  • Example: Given $18$ seats in row 1 ($a_1 = 18$) and $26$ seats in row 5 ($a_5 = 26$), solving $26 = 18 + 4d$ gives $d = 2$. The explicit formula is $a_n = 18 + 2(n - 1)$, and the 12th row has $a_{12} = 40$ seats.
  • Example: Given $a_n = 16 - 3(n - 1)$, substituting $n = 10$ gives a_{10} = -11$.\n* Recursive Formula:\n * Defines each term using operations on the previous term: a_1 = \text{first term}$and$a_n = a_{n-1} + d$for$n > 1$.
  • Example: For sequence $5, 7.5, 10, 12.5$, $a_1 = 5$ and $a_n = a_{n-1} + 2.5$ for n > 1$.\n * Example: For sequence 19, 13, 7, 1, -5$,$b_1 = 19$and$b_n = b_{n-1} - 6$for$n > 1$.
  • Conversion: For explicit formula $a_n = 16 - 3(n - 1)$, $a_1 = 16$ and the recursive definition is $a_n = a_{n-1} - 3$ for $n > 1$. For $a_n = 45 - 2(n - 1)$, the recursive definition is $a_n = a_{n-1} - 2$ for n > 1$.\n\n# Arithmetic Series and Sigma Notation\n\n* Arithmetic Series Formula:\n * Formula: S_n = \frac{n(a_1 + a_n)}{2}\n * Calculates the sum of n$terms given the first term$a_1$and last term$a_n$.
  • Example: For $12$ terms where $a_1 = 3$ and $a_{12} = 35$, S_{12} = \frac{12(3 + 35)}{2} = 228$.\n * Example: Sum of 1, 4, 7, 10, 13$($n = 5$):$S_5 = \frac{5(1 + 13)}{2} = 35$.
• Sigma Notation Formula:
  • Notation: $\sum_{i=1}^{n} a_i$
  • Steps to write a series in sigma notation:
  1. Find the number of terms $n$ using $a_n = a_1 + d(n - 1)$.
  2. Write and simplify the explicit formula $a_i = a_1 + d(i - 1)$.
  • Example: For series $2 + 9 + 16 + \dots + 79$, finding $79 = 2 + 7(n - 1)$ gives $n = 12$. The simplified explicit formula is $7i - 5$, represented as $\sum_{i=1}^{12} (7i - 5)$.
  • Example: To solve $\sum_{i=1}^{9} (2i - 6)$, find $n = 9$, $a_1 = 2(1) - 6 = -4$, and $a_9 = 2(9) - 6 = 12$. Applying the sum formula gives $S_9 = \frac{9(-4 + 12)}{2} = 36$.

Solving Equations and Inequalities by Graphing

• Memory Rhyme:
  Set each side to $y$, graph both on the grid; Where the lines intersect, the solution is hid!

• Linear and absolute value equations are solved graphically by setting each side equal to $y$, graphing both equations, and finding their point(s) of intersection:

  • For $-3x + 20 = 0.2x + 4$, graph $y = -3x + 20$ and $y = 0.2x + 4$. The lines intersect at $(5, 5)$, giving the solution $x = 5$.
  • For $|1x - 4| = \frac{1}{2}x + 1$, graph $y = |1x - 4|$ and $y = \frac{1}{2}x + 1$. The graphs intersect at $(2, 2)$ and $(10, 6)$, giving the solutions $x = 2$ and $x = 10$.

• Linear inequality word problems compare relative positions or distances over time:

  • For a motorcycle $40\,\text{mi}$ ahead traveling at $40\,\text{mph}$ and a car traveling at $60\,\text{mph}$, write the inequality $60x > 40x + 40$.
  • Solving yields $x > 2$, meaning the car will be ahead after $2\,\text{hours}$ (meeting at $(2, 120)$).

Arithmetic Sequences

• An arithmetic sequence is a sequence of numbers with a constant common difference $d$ (via addition or subtraction) between consecutive terms.

Explicit Formula

• Memory Rhyme:
  Start with term one, add $d$ to the mix, Multiply by $(n - 1)$ to find any term quick!
• Formula: $a<em>n = a</em>1 + d(n - 1)$
• Allows finding any term in a sequence without knowing the previous term.
• Example: Given $18$ seats in row 1 ($a<em>1 = 18$) and $26$ seats in row 5 ($a</em>5 = 26$), solving $26 = 18 + 4d$ gives $d = 2$. The explicit formula is $a<em>n = 18 + 2(n - 1)$, and the 12th row has $a</em>{12} = 40$ seats.
• Example: Given $a<em>n = 16 - 3(n - 1)$, substituting $n = 10$ gives a{10} = -11$.

Recursive Formula

• Memory Rhyme:
  To find where you're going, look right where you've been; Add d$to term$a_{n-1} to win!
• Defines each term using operations on the previous term: a1 = \text{first term}$and$an = a_{n-1} + d$for$n > 1$.
• Example: For sequence $5, 7.5, 10, 12.5$, $a<em>1 = 5$ and $a</em>n = a_{n-1} + 2.5$ for n > 1$.
• Example: For sequence 19, 13, 7, 1, -5$,$b1 = 19$and$bn = b_{n-1} - 6$for$n > 1$.
• Conversion: For explicit formula $a<em>n = 16 - 3(n - 1)$, $a</em>1 = 16$ and the recursive definition is $a<em>n = a</em>{n-1} - 3$ for $n > 1$. For $a<em>n = 45 - 2(n - 1)$, the recursive definition is $a</em>n = a_{n-1} - 2$ for n > 1$.

Arithmetic Series and Sigma Notation

Arithmetic Series Formula

• Memory Rhyme:
  First term plus last term, divide that by two; Multiply by n, and the sum comes to you!
• Formula: Sn = \frac{n(a1 + a_n)}{2}
• Calculates the sum of n$terms given the first term$a1$and last term$an$.
• Example: For $12$ terms where $a<em>1 = 3$ and $a</em>{12} = 35$, S_{12} = \frac{12(3 + 35)}{2} = 228$.
• Example: Sum of 1, 4, 7, 10, 13$($n = 5$):$S_5 = \frac{5(1 + 13)}{2} = 35$.

Sigma Notation Formula

• Memory Rhyme:
  Bottom is where you start, top is where you stop; Plug in the explicit rule, and sum to the top!
• Notation: $\sum<em>{i=1}^{n} a</em>i$
• Steps to write a series in sigma notation:
  1. Find the number of terms $n$ using $a<em>n = a</em>1 + d(n - 1)$.
  2. Write and simplify the explicit formula $a<em>i = a</em>1 + d(i - 1)$.
• Example: For series $2 + 9 + 16 + \dots + 79$, finding $79 = 2 + 7(n - 1)$ gives $n = 12$. The simplified explicit formula is $7i - 5$, represented as $\sum_{i=1}^{12} (7i - 5)$.
• Example: To solve $\sum<em>{i=1}^{9} (2i - 6)$, find $n = 9$, $a</em>1 = 2(1) - 6 = -4$, and $a<em>9 = 2(9) - 6 = 12$. Applying the sum formula gives $S</em>9 = \frac{9(-4 + 12)}{2} = 36$.