Question

Знайдіть суму членів арифметичної прогресї з 9 по 20 включно, якщо перший член дорівнює 5 різниця -2

Answer 1

А9=а1+8d=5–16=–11

2a1+11d
S12 = ------------ • 12=(2•(-11)+11•(-2))•6=
2

= -44•6 = –264

Answer 2

Calculating the Sum of an Arithmetic Progression

To find the sum of the members of an arithmetic progression from 9 to 20 inclusive, with the first member being 5 and the difference being -2, we can use the formula for the sum of an arithmetic progression.

The formula for the sum of the first n terms of an arithmetic progression is given by:

S_n = n/2 * (a_1 + a_n)

Where: - S_n is the sum of the first n terms - n is the number of terms - a_1 is the first term - a_n is the nth term

Applying the Formula

First, we need to find the number of terms in the given range. We can use the formula for the nth term of an arithmetic progression to find this.

The nth term of an arithmetic progression is given by:

a_n = a_1 + (n - 1)d

Where: - a_n is the nth term - a_1 is the first term - n is the number of terms - d is the common difference

Given: - a_1 = 5 - d = -2

We can use the formula to find the number of terms in the given range.

Finding the Number of Terms

Using the formula for the nth term, we can find the number of terms in the given range:

a_n = a_1 + (n - 1)d

Substituting the given values: 20 = 5 + (n - 1)(-2)

Solving for n: 20 = 5 - 2n + 2 20 - 5 = -2n + 2 15 = -2n + 2 13 = -2n n = -13 / -2 n = 13 / 2 n = 6.5

Since the number of terms must be a whole number, we take the integer part of 6.5, which is 6. Therefore, there are 6 terms in the given range.

Calculating the Sum

Now that we know there are 6 terms in the given range, we can use the formula for the sum of the first n terms to calculate the sum.

S_n = n/2 * (a_1 + a_n)

Substituting the given values: S_6 = 6/2 * (5 + a_6)

We need to find the value of a_6 using the formula for the nth term: a_6 = a_1 + (6 - 1)d a_6 = 5 + 5(-2) a_6 = 5 - 10 a_6 = -5

Substituting the value of a_6: S_6 = 3 * (5 + (-5)) S_6 = 3 * 0 S_6 = 0

Conclusion

The sum of the members of the arithmetic progression from 9 to 20 inclusive, with the first member being 5 and the difference being -2, is 0.