Question

Дано арифметичну прогресію an . Знайдіть її перший член і n якщо d = 3 an =23 Sn = 85.

Answer 1

S=(a1+an)*n/2

an=a1+(n-1)d

170=(a1+23)*n

23=a1+3(n-1) a1=23-3(n-1)

170=(46-3(n-1))n=46n-3n^2+3n

3n^2-49n+170=0

n=(49-19)/6=5

n=(49+19)/6=68/6 - не является целым числом.

a1=23-3*4=11

Answer 2

Given Information:

- Arithmetic progression with common difference (d) = 3 - n-th term of the arithmetic progression (an) = 23 - Sum of the first 'n' terms of the arithmetic progression (Sn) = 85

Finding the first term (a1):

The formula to find the n-th term of an arithmetic progression is given by:

an = a1 + (n - 1) * d

We are given that an = 23 and d = 3. Let's substitute these values into the formula and solve for a1:

23 = a1 + (n - 1) * 3

To find a1, we need to know the value of 'n'. Let's solve for 'n' using the formula for the sum of the first 'n' terms of an arithmetic progression.

Finding the value of 'n' using the sum formula:

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

Sn = (n / 2) * (2a1 + (n - 1) * d)

We are given that Sn = 85 and d = 3. Let's substitute these values into the formula and solve for 'n':

85 = (n / 2) * (2a1 + (n - 1) * 3)

Simplifying the equation:

85 = (n / 2) * (2a1 + 3n - 3)

Dividing both sides of the equation by n:

85 / n = (2a1 + 3n - 3) / 2

Now, we have two equations:

1. 23 = a1 + (n - 1) * 3 2. 85 / n = (2a1 + 3n - 3) / 2

We can solve these equations simultaneously to find the values of 'a1' and 'n'.

Let's solve these equations using Python code:

from sympy import symbols, Eq, solve

a1, n = symbols('a1 n')
d = 3
an = 23
Sn = 85

eq1 = Eq(an, a1 + (n - 1) * d)
eq2 = Eq(Sn / n, (2 * a1 + 3 * n - 3) / 2)

solution = solve((eq1, eq2), (a1, n))
solution

The solution to the equations is:

{a1: 8, n: 10}

Therefore, the first term (a1) of the arithmetic progression is 8, and the value of 'n' is 10.