Question

Сумма второго и восьмого членов равна 10, а сумма третьего и четырнадцатого 31. найдите разность прогрессии

Answer 1

$\left \{ {{a _{2} +a _{8}=10 } \atop {a _{3}+a _{14} =31 }} \right.$

$\left \{ {{a _{1}+d+a _{1}+7d=10 } \atop {a _{1} +2d+a _{1}+13d=31 }} \right.$

$-\left \{ {{2a _{1}+15d=31 } \atop {2a _{1}+8d=10 }} \right.$
___________
7d = 21
d = 3

Answer 2

Calculating the Difference of the Progression

To find the difference of the progression, we can use the given information about the sum of specific terms in the progression.

Given: - The sum of the second and eighth terms is 10. - The sum of the third and fourteenth terms is 31.

Let's denote the common difference of the arithmetic progression as d.

The sum of the second and eighth terms can be expressed as: 2nd term + 8th term = 10 Using the formula for the nth term of an arithmetic progression: a + (n-1)d + a + 7d = 10 2a + 8d = 10

The sum of the third and fourteenth terms can be expressed as: 3rd term + 14th term = 31 Using the formula for the nth term of an arithmetic progression: a + 2d + a + 13d = 31 2a + 15d = 31

Subtracting the equation for the sum of the second and eighth terms from the equation for the sum of the third and fourteenth terms will give us the value of the common difference, d.

Calculating the Common Difference: Subtracting the equation for the sum of the second and eighth terms from the equation for the sum of the third and fourteenth terms: (2a + 15d) - (2a + 8d) = 31 - 10 15d - 8d = 21 7d = 21 d = 3

So, the common difference of the arithmetic progression is 3.

Therefore, the difference of the progression is 3.