Answer :
To determine how many terms of the arithmetic series [tex]\(32 + 48 + 72 + \ldots\)[/tex] are needed to make the sum 665, let's follow these steps:
### Step 1: Identify the sequence characteristics
The given series is an arithmetic progression (AP) where:
- The first term [tex]\(a = 32\)[/tex]
- The second term is 48, so the common difference [tex]\(d = 48 - 32 = 16\)[/tex]
### Step 2: Use the sum formula of an arithmetic series
The formula for the sum of the first [tex]\(n\)[/tex] terms of an arithmetic sequence is given by:
[tex]\[ S_n = \frac{n}{2} \left(2a + (n-1)d\right) \][/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 3: Substitute the known values into the sum formula
We are given that the sum [tex]\(S_n = 665\)[/tex]. So we need:
[tex]\[ 665 = \frac{n}{2} \left(2(32) + (n-1)(16)\right) \][/tex]
### Step 4: Simplify the equation
[tex]\[ 665 = \frac{n}{2} \left(64 + 16n - 16\right) \][/tex]
Simplifying inside the parentheses:
[tex]\[ 665 = \frac{n}{2} \left(48 + 16n\right) \][/tex]
### Step 5: Further simplify and rearrange the equation
[tex]\[ 665 = \frac{n}{2} \times 48 + \frac{n}{2} \times 16n \][/tex]
[tex]\[ 665 = 24n + 8n^2 \][/tex]
### Step 6: Solve the quadratic equation
Now we need to solve the quadratic equation [tex]\(8n^2 + 24n - 665 = 0\)[/tex].
Using the quadratic formula [tex]\(n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 8\)[/tex], [tex]\(b = 24\)[/tex], and [tex]\(c = -665\)[/tex]:
[tex]\[ n = \frac{-24 \pm \sqrt{24^2 - 4 \cdot 8 \cdot (-665)}}{2 \cdot 8} \][/tex]
[tex]\[ n = \frac{-24 \pm \sqrt{576 + 21280}}{16} \][/tex]
[tex]\[ n = \frac{-24 \pm \sqrt{21856}}{16} \][/tex]
[tex]\[ \sqrt{21856} = 148 \quad \text{(rounded to 3 decimal places)} \][/tex]
[tex]\[ n = \frac{-24 \pm 148}{16} \][/tex]
### Step 7: Find the possible values for [tex]\(n\)[/tex]
[tex]\[ n_1 = \frac{124}{16} = 7.739 \][/tex]
[tex]\[ n_2 = \frac{-172}{16} = -10.739 \][/tex]
Since [tex]\(n\)[/tex] must be a positive value, we discard the negative solution, leaving:
[tex]\[ n \approx 7.739 \][/tex]
### Step 8: Conclusion
Since the number of terms must be a whole number, we round [tex]\(7.739\)[/tex] to the nearest whole number, giving [tex]\(n = 8\)[/tex]. However, the most accurate answer based on our calculation indicates that slightly fewer than 8 terms would sum to slightly less than 665. Therefore, it is acceptable to recognize that 7 terms are required, but the exact non-integer nature ([tex]\(n \approx 7.739\)[/tex]) indicates a refinement.
Thus, the number of terms required is approximately [tex]\(\boxed{7.739}\)[/tex].
### Step 1: Identify the sequence characteristics
The given series is an arithmetic progression (AP) where:
- The first term [tex]\(a = 32\)[/tex]
- The second term is 48, so the common difference [tex]\(d = 48 - 32 = 16\)[/tex]
### Step 2: Use the sum formula of an arithmetic series
The formula for the sum of the first [tex]\(n\)[/tex] terms of an arithmetic sequence is given by:
[tex]\[ S_n = \frac{n}{2} \left(2a + (n-1)d\right) \][/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 3: Substitute the known values into the sum formula
We are given that the sum [tex]\(S_n = 665\)[/tex]. So we need:
[tex]\[ 665 = \frac{n}{2} \left(2(32) + (n-1)(16)\right) \][/tex]
### Step 4: Simplify the equation
[tex]\[ 665 = \frac{n}{2} \left(64 + 16n - 16\right) \][/tex]
Simplifying inside the parentheses:
[tex]\[ 665 = \frac{n}{2} \left(48 + 16n\right) \][/tex]
### Step 5: Further simplify and rearrange the equation
[tex]\[ 665 = \frac{n}{2} \times 48 + \frac{n}{2} \times 16n \][/tex]
[tex]\[ 665 = 24n + 8n^2 \][/tex]
### Step 6: Solve the quadratic equation
Now we need to solve the quadratic equation [tex]\(8n^2 + 24n - 665 = 0\)[/tex].
Using the quadratic formula [tex]\(n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 8\)[/tex], [tex]\(b = 24\)[/tex], and [tex]\(c = -665\)[/tex]:
[tex]\[ n = \frac{-24 \pm \sqrt{24^2 - 4 \cdot 8 \cdot (-665)}}{2 \cdot 8} \][/tex]
[tex]\[ n = \frac{-24 \pm \sqrt{576 + 21280}}{16} \][/tex]
[tex]\[ n = \frac{-24 \pm \sqrt{21856}}{16} \][/tex]
[tex]\[ \sqrt{21856} = 148 \quad \text{(rounded to 3 decimal places)} \][/tex]
[tex]\[ n = \frac{-24 \pm 148}{16} \][/tex]
### Step 7: Find the possible values for [tex]\(n\)[/tex]
[tex]\[ n_1 = \frac{124}{16} = 7.739 \][/tex]
[tex]\[ n_2 = \frac{-172}{16} = -10.739 \][/tex]
Since [tex]\(n\)[/tex] must be a positive value, we discard the negative solution, leaving:
[tex]\[ n \approx 7.739 \][/tex]
### Step 8: Conclusion
Since the number of terms must be a whole number, we round [tex]\(7.739\)[/tex] to the nearest whole number, giving [tex]\(n = 8\)[/tex]. However, the most accurate answer based on our calculation indicates that slightly fewer than 8 terms would sum to slightly less than 665. Therefore, it is acceptable to recognize that 7 terms are required, but the exact non-integer nature ([tex]\(n \approx 7.739\)[/tex]) indicates a refinement.
Thus, the number of terms required is approximately [tex]\(\boxed{7.739}\)[/tex].
