Linear Equations
A linear equation in one variable is an equation that can be written in
the form ax+b=0, where a and b are constants and a \neq 0. The
goal is to find the value of x.
To solve a linear equation:
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Isolate the term with x.
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Divide by the coefficient of x.
Example 1. Solve 2x + 5 = 11.
Example 2. Solve 3(x-2) = 4x + 1.
Problem 1. Solve the following linear equations:
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5x - 7 = 18
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\frac{x}{3} + 2 = 7
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4(x+3) = 2(x-1)
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\frac{2x-1}{3} = \frac{x+5}{2}
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7 - (2x+3) = 5x - (x-4)
Arithmetic Sequences
An arithmetic sequence is a sequence of numbers where the difference
between consecutive terms is constant. This constant difference is
called the common difference, denoted by d.
Hence, an arithmetic sequence can be written as
a, a+d, a+2d, a+3d, \ldots. Here, a is the first term. The n-th
term, denoted by a_n, is given by a_n = a + (n-1)d.
Example 3. Consider the sequence 1, 3, 5, 7, \ldots. Here, a=1
and d=2. The terms are 1, 1+2, 1+2(2), 1+3(2), \ldots.
Example 4. Consider the sequence 10, 7, 4, 1, \ldots. Here,
a=10 and d=-3.
Example 5. Consider the sequence 2, 2.5, 3, 3.5, \ldots. Here,
a=2 and d=0.5.
Notice that the formula for the n-th term of an arithmetic sequence,
a_n = a + (n-1)d, is a linear equation in n. If we consider a_n as
the dependent variable and n as the independent variable, the common
difference d acts as the slope, and a-d acts as the y-intercept.
Now, let’s proceed with some practice problems.
Problem 2 (Useful property). Prove that
a-b, a, a+b are always in an arithmetic sequence for any a, b.
Problem 3. Find the 15th term of the arithmetic sequence
3, 8, 13, \ldots.
Problem 4. The 7th term of an arithmetic sequence is 23 and the
12th term is 43. Find the first term and the common difference.
Problem 5. If x+y, 2x, x+2y are in arithmetic sequence, find the
ratio x:y.
Problem 6. The sum of three consecutive terms of an arithmetic
sequence is 30, and their product is 750. Find the terms.
Problem 7 (Important property). If x, y, z
are three consecutive terms in an arithmetic sequence, then
y = \frac{x+z}{2}.
Problem 8. If x, y, z are in arithmetic sequence and x, 2y, 3z
are in arithmetic sequence, what is y-x?
Sum of an Arithmetic Sequence {#sum-of-an-arithmetic-sequence .unnumbered}
The sum of the first n terms of an arithmetic sequence, denoted by
S_n, can be found using the formula:
But it’s important to know from where this formula comes from. In its
essence, the formula is summing pairs of numbers of equal sums and
multiplying them by their number:
Problem 9. Find the sum S = 2 + 4 + 6 + \ldots + 100.
Problem 10 (Sum of odd numbers is a perfect square). Find the sum
S = 1 + 3 + 5 + \ldots + (2n-1).
Problem 11 (Alternating sum). Find the sum
S = 1 - 2 + 3 - 4 + \ldots + (-1)^{n+1} n.
Problem 12 (Challenge). *Find the sum:
Problem 13. Find the sum of the first 20 terms of the arithmetic sequence 5, 9, 13, \ldots.
Problem 14. The sum of the first n terms of an arithmetic
sequence is S_n = 3n^2 - 2n. Find the first term and the common
difference.
Problem 15. Find the sum of all integers between 100 and 500
(inclusive) that are divisible by 3.
Problem 16. Find the sum:
S = \frac{1}{1 \cdot 3} + \frac{1}{3 \cdot 5} + \frac{1}{5 \cdot 7} + \ldots + \frac{1}{19 \cdot 21}.
Problem 17. A clock strikes once at 1 o’clock, twice at 2 o’clock,
and so on. How many times does it strike in total during a 12-hour
period?
Problem 18. The first term of an arithmetic sequence is 5 and the
last term is 45. If the sum of the terms is 400, find the number of
terms and the common difference.