A kosarad üres
Arithmetic progressions
In practice one can often find sequences which can be defined as follows: the difference of any two consecutive terms is constant, that is,
\latex{a_{n+1}-a_n=d}.
These sequences are called arithmetic progressions.
DEFINITION: We say that the sequence \latex{\left(a_n\right)} is an arithmetic progression if there exists numbers \latex{a} and \latex{d} such that \latex{a_1 = a}, and for \latex{n \geq 1}, \latex{a_{n + 1} = a_n + d}.
For example, the sequence of positive odd numbers is an arithmetic progression.
\latex{1, 3, 5, 7, 9, 11, ..., 2n-1, ...}
\latex{a_1=1, a_{n+1}=a_n+2}
\latex{a_1=a} is the first term of the progression while \latex{d} is called the progression's difference.
It is easy to express the \latex{n^{th}} term of an arithmetic progression using the first term, \latex{n} and the difference:
\latex{a_n=a_1+\left(n-1\right)\times d}
We can prove the formula's validity by, for example, induction. It is true for \latex{n = 1}, \latex{a_1 = a_1}. Suppose it is true for \latex{n}, then
\latex{a_{n+1}=a_n+d=a_1+\left(n-1\right)\times d + d = a_1+n \times d}
Thus it is true for \latex{n + 1} as well.
Using the recursive definition, the terms of an arithmetic progression can be illustrated using the graph of the function \latex{f(x) = x + d}, \latex{x \in \R}. For example the progression with \latex{a = 1} and \latex{d = 2} can be seen in Figure 14.
\latex{y}
\latex{x}
\latex{1}
\latex{0}
\latex{1}
\latex{2}
\latex{3}
\latex{5}
\latex{7}
\latex{9}
\latex{y=x+2}
\latex{y=x}
Figure 14
Example 1
Compute the sum of the first \latex{n} terms of the arithmetic progression.
\latex{s_n=a_1+a_2+a_3+ \dots +a_n=} ?
Solution
Use the following simple fact:
\latex{a_1+a_n = a_2 + a_{n-1}=a_3+a_{n-2}=\dots= a_n+a_{1'}}
It follows that
\latex{2 \times s_n=n \times \left(a_1+a_n \right)},
\latex{s_n=\frac{a_1+a_n}{2}\times n}.
In the case when \latex{a} and \latex{d} are positive, the path leading to the result is illustrated by Figure 15.
\latex{a_n}
\latex{a_{n-1}}
\latex{a_{n-2}}
\latex{a_n}
\latex{a_1}
\latex{a_1}
\latex{a_2}
\latex{a_3}
Figure 15
Using the formula we have for \latex{a_n} we can express \latex{s_n} using \latex{a_1}, \latex{d} and \latex{n}:
\latex{s_n=\frac{2a_1+\left(n-1\right) \times d}{2}\times n}.
Example 2
Calculate the sum of all two-digit numbers which are divisible by \latex{3}.
Solution
The first two-digit number which is a multiple of \latex{3} is \latex{12} and then every third number is divisible by \latex{3}, the last summand is \latex{99}. We have to calculate the following sum:
\latex{12+15+18+\dots+96+99}.
This is an arithmetic progression with \latex{a_1 = 12} and \latex{d = 3}. The number of terms, \latex{n}, can be found by solving:
\latex{99=12+\left(n-1\right)\times 3}, thus \latex{n=30}.
Therefore the sum is
\latex{\frac{12+99}{2}\times 30=111 \times 15=1665}.
Example 3
The fifth term of an arithmetic progression is \latex{11} and the eighth is \latex{17}. Calculate the sum of the first \latex{10} terms of the progression.
Solution
By the conditions,
\latex{a_5=a+4d = 11} and \latex{a_8=a+7d=17}.
It follows that \latex{3d = 6}, therefore
\latex{d=2} and thus \latex{a=3}.
The first term of the progression is \latex{3}, its difference is \latex{2}. Then the sum of the first \latex{10} terms is
\latex{s_{10}=\frac{2 \times 3 + 9 \times 2}{2}\times 10 = 120}.
Example 4
We know the following about an arithmetic progression:
\latex{\begin{aligned}a_2+a_4+a_6=36\\a_2\times a_3 = 54\end{aligned}}
Find the first term and difference of the progression.
Solution
Using the first equality,
\latex{\begin{align*}3a+9d=36,\\a+3d=12,\\a=12-3d.\end{align*}}
Using the second equality,
\latex{\left(a+d\right) \times \left(a+2d\right)=54},
we can substitute the expression we have for \latex{a} into this equality:
\latex{\left(12-2d\right) \times \left(12-d\right)=54},
After rearranging it yields the quadratic equation
\latex{d^2-18d+45=0}.
Its roots are \latex{d_1 = 3}, \latex{d_2 = 15}, the corresponding values of \latex{a}: \latex{3} and \latex{–33}.
The first term and difference of the progression are either
\latex{\qquad a_1=3} and \latex{d=3},
or \latex{a_1=-33} and \latex{d=15}.
Both sequences satisfies the conditions.
Example 5
In an arithmetic progression, every term is a positive integer. We know that the progression’s second term is \latex{12} and the sum of the first \latex{9} terms is greater than \latex{200} and lesser than \latex{220}. Find the first term and difference of the progression.
Solution
Denote the first term by \latex{a}, the difference by \latex{d}. Since the second term is \latex{12}, \latex{a + d = 12}.
The sum of the first \latex{9} terms is
\latex{\frac{2a+8d}{2}\times 9=9 \times \left(a+4d\right)=9\times \left(12+3d\right)},
where we used that \latex{a+d=12}.
It is easier to factor out \latex{3} from the bracket. According to the condition, the sum is:
\latex{200 \lt 27 \times \left(d + 4\right) \lt 220}.
Since \latex{7 \times 27=189\lt 200} and \latex{9 \times 27 = 243\lt 220}:
\latex{7\lt d+4\lt9},
\latex{3 \lt d \lt 5}.
Since the terms of the sequence are integers, \latex{d} is also an integer, therefore
\latex{d=4} and \latex{a=8}.
The resulting progression satisfies the conditions:
\latex{a_2=a+d=12} and \latex{s_9=\frac{2a+8d}{2}\times9 = 216}.
Example 6
In an arithmetic progression, the sum of three consecutive terms is \latex{3}, while the sum of their cubes is \latex{15}. What is the difference of the progression?
Solution
Denote the three consecutive terms by \latex{x}, \latex{y}, \latex{z}. By the conditions,
\latex{\begin{align*}x+y+z=3,\\x^3+y^3+z^3=15,\\y-x=z-y.\end{align*}}
It follows from the third equation that \latex{x + z = 2y}, by substituting this into the first one we get \latex{y = 1}. Now the first two equations here become
\latex{x+z=2} and \latex{x^3+z^3=14}.
By expressing \latex{z} from the first and substituting it into the second we get the following after rearranging:
\latex{x^2-2x-1=0}.
It follows that \latex{x_1=1+\sqrt{2}}. \latex{x_2=1-\sqrt{2}}.
Thus the difference of the arithmetic progression can be either \latex{d_1= –\sqrt{2}} and \latex{d_2 = \sqrt{2}}.
It is easy to check that the resulting sequences satisfy the conditions.
Example 7
Prove that if \latex{0 \lt k \lt n} are integers and \latex{\left(a_n\right)} is an arithmetic progression, then
\latex{a_n=\frac{a_{n-k}+a_{n+k}}{2}}.
Solution
Express the \latex{\left(n–k\right)^{\text{th}}} and \latex{\left(n+k\right)^{\text{th}}} term using the known formula:
\latex{\begin{align*}a_{n-k}=a_1+\left(n-k-l\right) \times d,\\a_{n+k}=a_1+\left(n+k-l\right) \times d.\end{align*}}
By adding these two and dividing by \latex{2} we get
\latex{\frac{a_{n-k}+a_{n+k}}{2}=a_1+\left(n-1\right)\times d=a_n}.
The result can be briefly summarized as saying that any term of an arithmetic progression is the mean of any two terms located symmetrically about it. It is worth noting the result for \latex{k = 1}:
\latex{\frac{a_{n-1}+a_{n+1}}{2}}, if \latex{n\gt1} is an integer.
This means that for any three terms of the arithmetic progression the middle one is the mean of the two on the “sides”.
Example 8
The differences of consecutive terms in the sequence \latex{2;\, 5;\, 11;\, 20;\, \dots} form an arithmetic progression. Express the \latex{n^{\text{th}}} term of the sequence using \latex{n}.
Solution
Denote the terms of the sequence the usual way: \latex{a_1 = 2, a_2 = 5, a_3 = 11, a_4 = 20,\space\dots} By the condition \latex{a_2 - a_1 = 3, a_3 - a_2 = 6, a_4-a_3 = 9,\space\dots,} so for the \latex{n^{\text{th}}} term in the sequence of the differences:
\latex{a_{n+1}-a_n=3+\left(n-1\right)\times3}, in other words,
\latex{a_{n+1}=a_n+3+\left(n-1\right)\times 3}.
\latex{a_{n+1}=a_n+3+\left(n-1\right)\times 3}.
Write down the obtained equality for \latex{n = 1, 2,\space\dots ,\space n - 1} and sum the appropriate sides.
\latex{\begin{matrix*}a_2=a_1+3,\\a_3=a_2+3+1\times3,\\a_4=a_3+3+2 \times 3,\\\vdots\\a_n=a_{n-1}+3+\left(n-2\right)\times3.\end{matrix*}}
After adding and simplifying it follows that
\latex{\begin{align*}a_n=a_1+\left(n-1\right)\times 3 + 3 \times \left(1+2+\dots+n-2\right)=\\=2+3\times \left(1+2+\dots+n-1\right)=\frac{3}{2}\times n \times \left(n-1\right)+2.\end{align*}}
Example 9
Prove that if \latex{a}, \latex{b}, \latex{c} are three consecutive terms in an arithmetic progression, then \latex{a^2 + ab + b^2}, \latex{a^2 + ac + c^2}, \latex{b^2 + bc + c^2} are also three consecutive terms in some arithmetic progression.
Solution
By the condition \latex{b - a = c - b}. We show that the other three numbers also have this property.
\latex{\left(a^2+ac+c^2\right)-\left(a^2+ab+b^2\right)=c^2-b^2+a\times\left(c-b\right)=\left(c-b\right)\times \left(a+b+c\right)},
\latex{\left(b^2+bc+c^2\right)-\left(a^2+ac+c^2\right)=b^2-a^2+c\times\left(b-a\right)=\left(b-a\right)\times \left(a+b+c\right)}.
\latex{\left(b^2+bc+c^2\right)-\left(a^2+ac+c^2\right)=b^2-a^2+c\times\left(b-a\right)=\left(b-a\right)\times \left(a+b+c\right)}.
It follows from the condition that these differences are equal, that is, the proposition is true.
Exercises
{{exercise_number}}. Compute the sum of the positive integers which are not greater than \latex{1,000} and are divisible by \latex{3}.
{{exercise_number}}. The third term of an arithmetic progression is \latex{10}, the eighth term is \latex{30}. Which is the smallest \latex{n \gt 0} integer for which the sum of the progression's first n terms is at least \latex{1,000}?
{{exercise_number}}. The numbers \latex{a^2, b^2, c^2} are three consecutive terms of an arithmetic progression. Prove that there exists an arithmetic progression in which \latex{\frac{1}{b+c}, \frac{1}{a+c}, \frac{1}{a+b}} are three consecutive terms.
{{exercise_number}}. The following are known about the \latex{\left(a_n\right)} arithmetic progression:
- \latex{a_2+a_8=10\\a_3+a_{14}=31;}
- \latex{a_2+a_3+a_4+a_5=34,\\a_2 \times a_3=28;}
- \latex{{a_3}^2+{a_7}^2=122,\\a_1+a_7=4.}
{{exercise_number}}. Can \latex{1}, \latex{\sqrt{3}} and \latex{3} be three, not necessarily consecutive, terms of an arithmetic progression?
{{exercise_number}}. The length of the sides of a convex quadrilateral, in clockwise order, are \latex{a, b, c, d}. Prove that if \latex{a, b, d} and \latex{c} in this order are consecutive terms of an arithmetic progression, then the quadrilateral is tangential.
{{exercise_number}}. Express the following sum as simply as possible.
\latex{100^2-99^2+98^2-97^2+\dots+2^2-1^2}.
{{exercise_number}}. An object, which is in free fall, moves \latex{4.9\,m} in the first second, and in every later second \latex{9.8\,m} more than in the last second before. How long does it take the object to fall from a height of \latex{4,410\,m}?
{{exercise_number}}. How many times does a bell toll during one day (\latex{24} hours) if it only strikes on the hour. Also, for example at \latex{4} pm, it strikes \latex{4} times.
{{exercise_number}}. How many such points \latex{P\left(x; y\right)} are there in the plane where both its coordinates are integers and they satisfying the inequality
\latex{|x|+|y|\leq 10}?
