Question

Срочно надо! Найдите сумму двенадцати первых членов арифметической прогрессии -5; -2; 1; ... .

Answer 1

Решение
d = - 2 - (-5) = 3
Sn = (a₁ + an)*12 / 2
a₁₂ = - 5 + 11*3 = 28
Sn = (-5 + 28)*6 = 23*6 = 138

Answer 2

Calculating the Sum of the First Twelve Terms of an Arithmetic Progression

To find the sum of the first twelve terms of the arithmetic progression -5, -2, 1, ..., we can use the formula for the sum of an arithmetic series:

Sum of an Arithmetic Series: The sum of the first n terms of an arithmetic series is given by the formula: \[ S_n = \frac{n}{2}(a_1 + a_n) \]

Where: - \( S_n \) = sum of the first n terms - \( n \) = number of terms - \( a_1 \) = first term - \( a_n \) = nth term

In this case, we have: - \( n = 12 \) (as we want to find the sum of the first twelve terms) - \( a_1 = -5 \) (the first term) - \( a_{12} = ? \) (the twelfth term)

Finding the Twelfth Term of the Arithmetic Progression

To find the twelfth term of the arithmetic progression, we can use the formula for the nth term of an arithmetic sequence: \[ a_n = a_1 + (n - 1)d \]

Where: - \( a_n \) = nth term - \( a_1 \) = first term - \( n \) = term number - \( d \) = common difference

In this case, the common difference is 3 (since each term is 3 greater than the previous term).

Using the formula, we can find the twelfth term: \[ a_{12} = -5 + (12 - 1) \times 3 \] \[ a_{12} = -5 + 33 \] \[ a_{12} = 28 \]

Calculating the Sum

Now that we have the first term \( a_1 = -5 \) and the twelfth term \( a_{12} = 28 \), we can calculate the sum of the first twelve terms using the formula for the sum of an arithmetic series: \[ S_{12} = \frac{12}{2}(-5 + 28) \] \[ S_{12} = 6 \times 23 \] \[ S_{12} = 138 \]

Therefore, the sum of the first twelve terms of the arithmetic progression -5, -2, 1, ... is 138.