To determine the number of terms in the arithmetic sequence needed to achieve a sum of 3596, given that the first term is 5 and the common difference is 2, follow these steps:
### Step 1: Formula for the Sum of an Arithmetic Sequence
The sum of the first [tex]\(n\)[/tex] terms of an arithmetic sequence can be represented by the formula:
[tex]\[ S_n = \frac{n}{2} [2a + (n-1)d] \][/tex]
where:
- [tex]\(S_n\)[/tex] is the sum of the first [tex]\(n\)[/tex] terms,
- [tex]\(a\)[/tex] is the first term,
- [tex]\(d\)[/tex] is the common difference,
- [tex]\(n\)[/tex] is the number of terms.
### Step 2: Substitute the Given Values
We know the following:
- [tex]\(a = 5\)[/tex]
- [tex]\(d = 2\)[/tex]
- [tex]\(S_n = 3596\)[/tex]
Substitute these values into the formula:
[tex]\[ 3596 = \frac{n}{2} [2(5) + (n-1)(2)] \][/tex]
### Step 3: Simplify the Equation
First, expand the bracket and simplify:
[tex]\[ 3596 = \frac{n}{2} [10 + 2n - 2] \][/tex]
[tex]\[ 3596 = \frac{n}{2} [2n + 8] \][/tex]
Then, distribute [tex]\(\frac{n}{2}\)[/tex]:
[tex]\[ 3596 = n \left(\frac{2n + 8}{2}\right) \][/tex]
[tex]\[ 3596 = n(n + 4) \][/tex]
This simplifies to a quadratic equation:
[tex]\[ n^2 + 4n - 3596 = 0 \][/tex]
### Step 4: Solve the Quadratic Equation
To solve the quadratic equation [tex]\( n^2 + 4n - 3596 = 0 \)[/tex], we use the quadratic formula:
[tex]\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
where [tex]\(a = 1\)[/tex], [tex]\(b = 4\)[/tex], and [tex]\(c = -3596\)[/tex]. Substitute these values into the formula:
[tex]\[ n = \frac{-4 \pm \sqrt{4^2 - 4(1)(-3596)}}{2(1)} \][/tex]
[tex]\[ n = \frac{-4 \pm \sqrt{16 + 14384}}{2} \][/tex]
[tex]\[ n = \frac{-4 \pm \sqrt{14400}}{2} \][/tex]
[tex]\[ n = \frac{-4 \pm 120}{2} \][/tex]
This results in two possible solutions:
[tex]\[ n_1 = \frac{116}{2} = 58 \][/tex]
[tex]\[ n_2 = \frac{-124}{2} = -62 \][/tex]
### Step 5: Interpret the Solutions
Since the number of terms [tex]\(n\)[/tex] must be a positive integer, we discard the negative solution. Therefore, the number of terms required is:
[tex]\[ n = 58 \][/tex]
So, the arithmetic sequence needs 58 terms to reach a sum of 3596.
