Question

Известно,что сумма второго,шестого и девятнадцатого членов арифметической прогрессии равна 39 .Определите номер члена этой прогрессии при сложении которого с третьим,седьмым и десятым членом в результате получается 52

Answer 1

$a_2+a_6+a_{19}=39\; \; ,\; \; \; a_{n}+a_3+a_7+a_{10}=52\; \; ,\; \; n=?\\\\ (a_1+d)+(a_1+5d)+(a_1+18d)=39\\\\3a_1+24d=39\; |:3\; \; \; \to \; \; \; a_1+8d=13\\\\a_1=13-8d\\\\\\(a_1+d(n-1))+(a_1+2d)+(a_1+6d)+(a_1+9d)=52\\\\4a_1+16d+dn=52\\\\4(13-8d)+16d+dn=52\\\\52-32d+16d+dn=52\\\\-16d+dn=0\\\\dn=16d\\\\n=\frac{16d}{d}\\\\n=16$

Answer 2

Problem Analysis

We are given that the sum of the second, sixth, and nineteenth terms of an arithmetic progression is 39. We need to determine the number of the term in this progression such that when it is added to the third, seventh, and tenth terms, the sum is 52.

Solution

Let's denote the first term of the arithmetic progression as a and the common difference as d. The formula for the nth term of an arithmetic progression is given by:

T(n) = a + (n - 1)d

We are given that the sum of the second, sixth, and nineteenth terms is 39. Using the formula for the sum of an arithmetic progression, we can write:

T(2) + T(6) + T(19) = 39

Substituting the formula for the nth term, we get:

(a + d) + (a + 5d) + (a + 18d) = 39

Simplifying the equation, we have:

3a + 24d = 39 Next, we need to find the number of the term in the progression such that when it is added to the third, seventh, and tenth terms, the sum is 52. Using the formula for the sum of an arithmetic progression, we can write:

T(n) + T(n + 4) + T(n + 9) = 52

Substituting the formula for the nth term, we get:

(a + (n - 1)d) + (a + (n + 3)d) + (a + (n + 8)d) = 52

Simplifying the equation, we have:

3a + 3n + 21d = 52 We now have a system of two equations with two variables (a, d) and (n). We can solve this system of equations to find the values of a, d, and n.

Solving the System of Equations

To solve the system of equations, we can use the method of substitution. We can rearrange equation to solve for a in terms of d:

a = (39 - 24d) / 3

Substituting this value of a into equation we get:

(39 - 24d) / 3 + 3n + 21d = 52

Multiplying through by 3 to eliminate the fraction, we have:

39 - 8d + 9n + 63d = 156

Simplifying the equation, we get:

9n + 55d = 117 Now we have a single equation with two variables (n and d). We can solve this equation to find the value of n.

To solve equation we can rearrange it to solve for n in terms of d:

n = (117 - 55d) / 9

Since n represents the number of a term in the arithmetic progression, it must be a positive integer. We can substitute different values of d and check if the resulting n is a positive integer.

Let's substitute some values of d and calculate the corresponding n:

- For d = 1: n = (117 - 55) / 9 = 62 / 9 ≈ 6.89

- For d = 2: n = (117 - 2 * 55) / 9 = 7

- For d = 3: n = (117 - 3 * 55) / 9 = -4.33

From the above calculations, we can see that d = 2 gives us a positive integer value for n. Therefore, the number of the term in the arithmetic progression that satisfies the given conditions is n = 7.

Answer

The number of the term in the arithmetic progression that, when added to the third, seventh, and tenth terms, results in a sum of 52 is 7.