Find those arithmetic sequence, which contain all powers of thier terms
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In mathematics, an arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with common difference of 2.
If the initial term of an arithmetic progression is

and the common difference of successive members is d, then the nth term of the sequence (

) is given by:

,
and in general

.
A finite portion of an arithmetic progression is called a finite arithmetic progression and sometimes just called an arithmetic progression. The sum of a finite arithmetic progression is called an arithmetic series.
The behavior of the arithmetic progression depends on the common difference d. If the common difference is:
positive, then the members (terms) will grow towards positive infinity;
negative, then the members (terms) will grow towards negative infinity.
Sum
2
+
5
+
8
+
11
+
14
=
40
14
+
11
+
8
+
5
+
2
=
40
16
+
16
+
16
+
16
+
16
=
80
Computation of the sum 2 + 5 + 8 + 11 + 14. When the sequence is reversed and added to itself term by term, the resulting sequence has a single repeated value in it, equal to the sum of the first and last numbers (2 + 14 = 16). Thus 16 × 5 = 80 is twice the sum.
The sum of the members of a finite arithmetic progression is called an arithmetic series. For example, consider the sum:
This sum can be found quickly by taking the number n of terms being added (here 5), multiplying by the sum of the first and last number in the progression (here 2 + 14 = 16), and dividing by 2:
In the case above, this gives the equation:
This formula works for any real numbers and . For example:
Derivation

Animated proof for the formula giving the sum of the first integers 1+2+...+n.
To derive the above formula, begin by expressing the arithmetic series in two different ways:
Adding both sides of the two equations, all terms involving d cancel:
Dividing both sides by 2 produces a common form of the equation:
An alternate form results from re-inserting the substitution: :
Furthermore, the mean value of the series can be calculated via: :
In 499 AD Aryabhata, a prominent mathematician-astronomer from the classical age of Indian mathematics and Indian astronomy, gave this method in the Aryabhatiya (section 2.18).
If the initial term of an arithmetic progression is

and the common difference of successive members is d, then the nth term of the sequence (

) is given by:

,
and in general

.
A finite portion of an arithmetic progression is called a finite arithmetic progression and sometimes just called an arithmetic progression. The sum of a finite arithmetic progression is called an arithmetic series.
The behavior of the arithmetic progression depends on the common difference d. If the common difference is:
positive, then the members (terms) will grow towards positive infinity;
negative, then the members (terms) will grow towards negative infinity.
Sum
2
+
5
+
8
+
11
+
14
=
40
14
+
11
+
8
+
5
+
2
=
40
16
+
16
+
16
+
16
+
16
=
80
Computation of the sum 2 + 5 + 8 + 11 + 14. When the sequence is reversed and added to itself term by term, the resulting sequence has a single repeated value in it, equal to the sum of the first and last numbers (2 + 14 = 16). Thus 16 × 5 = 80 is twice the sum.
The sum of the members of a finite arithmetic progression is called an arithmetic series. For example, consider the sum:
This sum can be found quickly by taking the number n of terms being added (here 5), multiplying by the sum of the first and last number in the progression (here 2 + 14 = 16), and dividing by 2:
In the case above, this gives the equation:
This formula works for any real numbers and . For example:
Derivation

Animated proof for the formula giving the sum of the first integers 1+2+...+n.
To derive the above formula, begin by expressing the arithmetic series in two different ways:
Adding both sides of the two equations, all terms involving d cancel:
Dividing both sides by 2 produces a common form of the equation:
An alternate form results from re-inserting the substitution: :
Furthermore, the mean value of the series can be calculated via: :
In 499 AD Aryabhata, a prominent mathematician-astronomer from the classical age of Indian mathematics and Indian astronomy, gave this method in the Aryabhatiya (section 2.18).
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Answer:
check whether the sequence of powers of thike всё подняться ив рошеля ав an aritmetic sequence of not. If it is an anthmetic sequence find its comm diffences.
Is lol a term of the arithmetic sequence
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