34 ! 3) Определите, является ли число -12 элементом арифметической прогрессии, в которой первый член равен

Velvet

25

22. Let"s start by solving problem 3.

To determine if -12 is an element of an arithmetic progression, we need to check if it satisfies the condition of an arithmetic sequence. An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant.

In this problem, the first term (a) is given as 14, and the difference (d) is given as -2.

To find out if -12 is part of this arithmetic progression, we can use the formula for the nth term of an arithmetic sequence:

\[a_n = a + (n - 1) \cdot d\]

Substituting the given values into the formula, we have:

\[a_n = 14 + (3 - 1) \cdot (-2)\]

Simplifying the equation, we get:

\[a_n = 14 + 2 \cdot (-2)\]

\[a_n = 14 - 4\]

\[a_n = 10\]

So, the 3rd term of this arithmetic progression is 10. Since -12 does not equal 10, we can conclude that -12 is not an element of this arithmetic progression.

Now, let"s move on to problem 4.

In this problem, the 5th term (a_5) is given as 14, and the 9th term (a_9) is given as ?.

We need to find the first term (a) and the difference (d) of this arithmetic progression.

We can use the formula for the nth term of an arithmetic sequence to find the first term:

\[a_n = a + (n - 1) \cdot d\]

For the 5th term, we have:

\[a_5 = a + (5 - 1) \cdot d\]

Substituting the given values into the equation, we have:

\[14 = a + 4d \quad \text{(equation 1)}\]

Similarly, for the 9th term, we have:

\[a_9 = a + (9 - 1) \cdot d\]

Substituting the given values into the equation, we have:

\[a_9 = a + 8d \quad \text{(equation 2)}\]

Now, we have a system of two equations (equations 1 and 2) with two variables (a and d). We can solve this system of equations to find the values of a and d.

Let"s subtract equation 1 from equation 2:

\[(a_9 - a_5) = (a + 8d) - (a + 4d)\]

\[14 - 14 = a + 8d - a - 4d\]

\[0 = 4d\]

Since 4d = 0, we can conclude that d = 0.

Substituting this value of d back into equation 1, we have:

\[14 = a + 4 \cdot 0\]

\[14 = a\]

Therefore, the first term (a) of the arithmetic progression is 14 and the difference (d) is 0.

To summarize,
- In problem 3, -12 is not an element of the arithmetic progression with the first term 14 and the difference -2.
- In problem 4, the first term of the arithmetic progression is 14 and the difference is 0.

Please let me know if you have any further questions or need clarification on any part of the explanation.