Answer :
Let's solve this problem step-by-step to find the number of terms in the arithmetic sequence.
1. Identify the given values:
- Sum of the arithmetic sequence, [tex]\( S = 55 \)[/tex]
- First term, [tex]\( a = 5 \)[/tex]
- Common difference, [tex]\( d = 3 \)[/tex]
2. Understand the formula for the sum of an arithmetic sequence:
[tex]\[
S = \frac{n}{2} (2a + (n-1)d)
\][/tex]
Where:
- [tex]\( S \)[/tex] is the sum of the sequence,
- [tex]\( n \)[/tex] is the number of terms,
- [tex]\( a \)[/tex] is the first term, and
- [tex]\( d \)[/tex] is the common difference.
3. Plug the values into the formula:
[tex]\[
55 = \frac{n}{2} \left(2 \times 5 + (n-1) \times 3\right)
\][/tex]
Simplify inside the parentheses:
[tex]\[
55 = \frac{n}{2} (10 + 3n - 3)
\][/tex]
Which simplifies further to:
[tex]\[
55 = \frac{n}{2} (7 + 3n)
\][/tex]
4. Remove the fraction by multiplying both sides by 2:
[tex]\[
110 = n (7 + 3n)
\][/tex]
5. Expand and rearrange the equation:
[tex]\[
110 = 7n + 3n^2
\][/tex]
Rearrange terms to form a quadratic equation:
[tex]\[
3n^2 + 7n - 110 = 0
\][/tex]
6. Solve the quadratic equation using the quadratic formula:
The quadratic formula is:
[tex]\[
n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Substitute [tex]\( a = 3 \)[/tex], [tex]\( b = 7 \)[/tex], [tex]\( c = -110 \)[/tex]:
- [tex]\( b^2 - 4ac = 7^2 - 4 \times 3 \times (-110) \)[/tex]
- Calculate the discriminant: [tex]\( 49 + 1320 = 1369 \)[/tex]
- Take the square root of the discriminant: [tex]\( \sqrt{1369} = 37 \)[/tex]
7. Find the two possible values for [tex]\( n \)[/tex]:
- First solution:
[tex]\[
n = \frac{-7 + 37}{6} = \frac{30}{6} = 5
\][/tex]
- Second solution:
[tex]\[
n = \frac{-7 - 37}{6} = \frac{-44}{6} = -7.333...
\][/tex]
Since the number of terms [tex]\( n \)[/tex] must be a positive integer, the valid solution is [tex]\( n = 5 \)[/tex].
Therefore, the number of terms in the arithmetic sequence is 5.
1. Identify the given values:
- Sum of the arithmetic sequence, [tex]\( S = 55 \)[/tex]
- First term, [tex]\( a = 5 \)[/tex]
- Common difference, [tex]\( d = 3 \)[/tex]
2. Understand the formula for the sum of an arithmetic sequence:
[tex]\[
S = \frac{n}{2} (2a + (n-1)d)
\][/tex]
Where:
- [tex]\( S \)[/tex] is the sum of the sequence,
- [tex]\( n \)[/tex] is the number of terms,
- [tex]\( a \)[/tex] is the first term, and
- [tex]\( d \)[/tex] is the common difference.
3. Plug the values into the formula:
[tex]\[
55 = \frac{n}{2} \left(2 \times 5 + (n-1) \times 3\right)
\][/tex]
Simplify inside the parentheses:
[tex]\[
55 = \frac{n}{2} (10 + 3n - 3)
\][/tex]
Which simplifies further to:
[tex]\[
55 = \frac{n}{2} (7 + 3n)
\][/tex]
4. Remove the fraction by multiplying both sides by 2:
[tex]\[
110 = n (7 + 3n)
\][/tex]
5. Expand and rearrange the equation:
[tex]\[
110 = 7n + 3n^2
\][/tex]
Rearrange terms to form a quadratic equation:
[tex]\[
3n^2 + 7n - 110 = 0
\][/tex]
6. Solve the quadratic equation using the quadratic formula:
The quadratic formula is:
[tex]\[
n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Substitute [tex]\( a = 3 \)[/tex], [tex]\( b = 7 \)[/tex], [tex]\( c = -110 \)[/tex]:
- [tex]\( b^2 - 4ac = 7^2 - 4 \times 3 \times (-110) \)[/tex]
- Calculate the discriminant: [tex]\( 49 + 1320 = 1369 \)[/tex]
- Take the square root of the discriminant: [tex]\( \sqrt{1369} = 37 \)[/tex]
7. Find the two possible values for [tex]\( n \)[/tex]:
- First solution:
[tex]\[
n = \frac{-7 + 37}{6} = \frac{30}{6} = 5
\][/tex]
- Second solution:
[tex]\[
n = \frac{-7 - 37}{6} = \frac{-44}{6} = -7.333...
\][/tex]
Since the number of terms [tex]\( n \)[/tex] must be a positive integer, the valid solution is [tex]\( n = 5 \)[/tex].
Therefore, the number of terms in the arithmetic sequence is 5.
