Question

Является ли число -103 членом арифметической прогрессии, первый член которой равен 30, а пятый равен 2? Если да, то определите номер этого члена.

Answer 1

Ответ:

a1 = 30

a5 = 2

n ∈ N

a5 = a1 + d(n-1)

2 = 30 + 4d

4d = -28

d = -7

a1 + d(n-1) = -103

30 -7n - 7 = -103

-7n = -66

n = -9.4 - посторонний (n ∈ N)

-103 Не является  членом прогрессии

Answer 2

Checking if -103 is a Member of an Arithmetic Progression

To determine if -103 is a member of an arithmetic progression with the first term as 30 and the fifth term as 2, we can use the formula for the nth term of an arithmetic progression:

Formula for the nth term of an arithmetic progression: The nth term of an arithmetic progression is given by the formula: \[a_n = a_1 + (n-1)d\] where: - \(a_n\) = nth term - \(a_1\) = first term - \(n\) = position of the term - \(d\) = common difference

Calculating the nth Term

Using the given information: - First term (\(a_1\)) = 30 - Fifth term (\(a_5\)) = 2

We can use the formula to find the common difference (\(d\)): \[a_5 = a_1 + (5-1)d\] \[2 = 30 + 4d\] \[4d = -28\] \[d = -7\]

Now that we have the common difference, we can find the nth term for -103: \[-103 = 30 + (n-1)(-7)\]

Solving for the Position of the Term

Solving for \(n\): \[-103 = 30 - 7n + 7\] \[-133 = -7n\] \[n = 19\]

Conclusion: The number -103 is the 19th term of the given arithmetic progression.