To find the sum of the arithmetic sequence from 45 to 144, we need to follow these steps:
1. Identify the first term and the last term of the sequence.
- The first term ([tex]\(a\)[/tex]) is 45.
- The last term ([tex]\(l\)[/tex]) is 144.
2. Determine the number of terms in the sequence.
- To find the number of terms ([tex]\(n\)[/tex]), we use the formula:
[tex]\[
n = l - a + 1
\][/tex]
- Here, [tex]\(a = 45\)[/tex] and [tex]\(l = 144\)[/tex], so:
[tex]\[
n = 144 - 45 + 1 = 100
\][/tex]
3. Use the formula for the sum of an arithmetic sequence.
- The sum [tex]\(S\)[/tex] of an arithmetic sequence is given by the formula:
[tex]\[
S = \frac{n}{2} \times (a + l)
\][/tex]
- Substituting the values, we get:
[tex]\[
S = \frac{100}{2} \times (45 + 144)
\][/tex]
4. Calculate the sum.
- First, calculate the sum inside the parentheses:
[tex]\[
45 + 144 = 189
\][/tex]
- Then, multiply by [tex]\(\frac{n}{2}\)[/tex]:
[tex]\[
S = 50 \times 189 = 9450
\][/tex]
Therefore, the sum of the sequence from 45 to 144 is [tex]\(9450\)[/tex].
