Question

В арифметической прогрессии Сумма первых 9 членов равна 60 вычислить пятый член прогрессии Нужно решение

Answer 1

Сумму девяти членов арифметической прогрессии можно вычислить по формуле:
$S _{9} = \frac{(a _{1}+a _{9} )*9 }{2}$
2S₉ = (a₁ + a₉)*9
120 = (a₁ + a₁ + 8d)* 9
2a₁ + 8d = 120 : 9
2(a₁ + 4d) = 120 : 9
$a _{1}+ 4d = \frac{120}{9*2} = 6 \frac{2}{3}$
a₅ = a₁ + 4d = $6 \frac{2}{3}$

Answer 2

Arithmetical Progression and the Sum of the First 9 Terms

In an arithmetic progression, the terms increase or decrease by a constant difference. Let's denote the first term as \(a_1\) and the common difference as \(d\). The formula to find the sum of the first \(n\) terms of an arithmetic progression is:

\[S_n = \frac{n}{2}(2a_1 + (n-1)d)\]

Given that the sum of the first 9 terms is 60, we can set up the equation:

\[60 = \frac{9}{2}(2a_1 + 8d)\]

Now, we can solve for \(a_1\) and \(d\) using this equation.

Solving for the Fifth Term

To find the fifth term of the arithmetic progression, we need to know the value of \(a_1\) and \(d\). Once we have these values, we can use the formula:

\[a_n = a_1 + (n-1)d\]

Substituting the values of \(a_1\) and \(d\) into this formula will give us the fifth term.

Let's calculate the values step by step.

Step 1: Finding \(a_1\) and \(d\)

Using the equation \(60 = \frac{9}{2}(2a_1 + 8d)\), we can solve for \(a_1\) and \(d\).

From the equation, we can see that \(n = 9\), so we have:

\[60 = \frac{9}{2}(2a_1 + 8d)\]

Simplifying the equation, we get:

\[120 = 9(2a_1 + 8d)\]

Dividing both sides by 9, we have:

\[13.33 \approx 2a_1 + 8d\]

Step 2: Finding the Fifth Term

Now that we have the values of \(a_1\) and \(d\), we can use the formula \(a_n = a_1 + (n-1)d\) to find the fifth term.

Substituting \(n = 5\) into the formula, we get:

\[a_5 = a_1 + (5-1)d\]

Plugging in the values of \(a_1\) and \(d\) that we found in Step 1, we have:

\[a_5 = 13.33 + (5-1)d\]

Simplifying further, we get:

\[a_5 = 13.33 + 4d\]

Therefore, the fifth term of the arithmetic progression is \(13.33 + 4d\).

Final Answer

To find the exact value of the fifth term, we need to know the value of \(d\). Unfortunately, the value of \(d\) is not provided in the question. Without the value of \(d\), we cannot calculate the exact fifth term of the arithmetic progression.

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