Answer :
Certainly! Let's break down the solution step by step for the arithmetic progression (AP) problem.
### Given:
- The third term ([tex]\(a_3\)[/tex]) is [tex]\(-40\)[/tex].
- The thirteenth term ([tex]\(a_{13}\)[/tex]) is [tex]\(10\)[/tex].
### To Find:
1. The first term ([tex]\(a_1\)[/tex]).
2. The common difference ([tex]\(d\)[/tex]).
3. The 30th term ([tex]\(a_{30}\)[/tex]).
4. Which term in the sequence equals [tex]\(35\)[/tex].
### Step-by-Step Solution:
1. Find the Common Difference ([tex]\(d\)[/tex]):
In an AP, the [tex]\(n^{\text{th}}\)[/tex] term is given by the formula:
[tex]\[ a_n = a_1 + (n-1) \times d \][/tex]
- For the third term ([tex]\(a_3\)[/tex]):
[tex]\[ a_3 = a_1 + 2d = -40 \][/tex]
- For the thirteenth term ([tex]\(a_{13}\)[/tex]):
[tex]\[ a_{13} = a_1 + 12d = 10 \][/tex]
Subtract the equation for [tex]\(a_3\)[/tex] from the equation for [tex]\(a_{13}\)[/tex]:
[tex]\[
(a_1 + 12d) - (a_1 + 2d) = 10 - (-40)
\][/tex]
[tex]\[
10d = 50
\][/tex]
[tex]\[
d = 5
\][/tex]
2. Find the First Term ([tex]\(a_1\)[/tex]):
Using the equation for the third term:
[tex]\[ a_3 = a_1 + 2d = -40 \][/tex]
Substitute [tex]\(d = 5\)[/tex] into the equation:
[tex]\[ a_1 + 2 \times 5 = -40 \][/tex]
[tex]\[ a_1 + 10 = -40 \][/tex]
[tex]\[ a_1 = -40 - 10 \][/tex]
[tex]\[ a_1 = -50 \][/tex]
3. Find the 30th term ([tex]\(a_{30}\)[/tex]):
Substitute [tex]\(n = 30\)[/tex] into the formula:
[tex]\[ a_{30} = a_1 + 29d \][/tex]
[tex]\[ a_{30} = -50 + 29 \times 5 \][/tex]
[tex]\[ a_{30} = -50 + 145 \][/tex]
[tex]\[ a_{30} = 95 \][/tex]
4. Determine which term equals 35:
We need to find [tex]\(n\)[/tex] such that:
[tex]\[ a_n = a_1 + (n-1) \times d = 35 \][/tex]
Substitute the known values:
[tex]\[ 35 = -50 + (n-1) \times 5 \][/tex]
[tex]\[ 35 + 50 = (n-1) \times 5 \][/tex]
[tex]\[ 85 = (n-1) \times 5 \][/tex]
[tex]\[ n-1 = \frac{85}{5} \][/tex]
[tex]\[ n-1 = 17 \][/tex]
[tex]\[ n = 18 \][/tex]
### Conclusion:
- The first term ([tex]\(a_1\)[/tex]) is [tex]\(-50\)[/tex].
- The common difference ([tex]\(d\)[/tex]) is [tex]\(5\)[/tex].
- The 30th term ([tex]\(a_{30}\)[/tex]) is [tex]\(95\)[/tex].
- The term that equals [tex]\(35\)[/tex] is the 18th term.
### Given:
- The third term ([tex]\(a_3\)[/tex]) is [tex]\(-40\)[/tex].
- The thirteenth term ([tex]\(a_{13}\)[/tex]) is [tex]\(10\)[/tex].
### To Find:
1. The first term ([tex]\(a_1\)[/tex]).
2. The common difference ([tex]\(d\)[/tex]).
3. The 30th term ([tex]\(a_{30}\)[/tex]).
4. Which term in the sequence equals [tex]\(35\)[/tex].
### Step-by-Step Solution:
1. Find the Common Difference ([tex]\(d\)[/tex]):
In an AP, the [tex]\(n^{\text{th}}\)[/tex] term is given by the formula:
[tex]\[ a_n = a_1 + (n-1) \times d \][/tex]
- For the third term ([tex]\(a_3\)[/tex]):
[tex]\[ a_3 = a_1 + 2d = -40 \][/tex]
- For the thirteenth term ([tex]\(a_{13}\)[/tex]):
[tex]\[ a_{13} = a_1 + 12d = 10 \][/tex]
Subtract the equation for [tex]\(a_3\)[/tex] from the equation for [tex]\(a_{13}\)[/tex]:
[tex]\[
(a_1 + 12d) - (a_1 + 2d) = 10 - (-40)
\][/tex]
[tex]\[
10d = 50
\][/tex]
[tex]\[
d = 5
\][/tex]
2. Find the First Term ([tex]\(a_1\)[/tex]):
Using the equation for the third term:
[tex]\[ a_3 = a_1 + 2d = -40 \][/tex]
Substitute [tex]\(d = 5\)[/tex] into the equation:
[tex]\[ a_1 + 2 \times 5 = -40 \][/tex]
[tex]\[ a_1 + 10 = -40 \][/tex]
[tex]\[ a_1 = -40 - 10 \][/tex]
[tex]\[ a_1 = -50 \][/tex]
3. Find the 30th term ([tex]\(a_{30}\)[/tex]):
Substitute [tex]\(n = 30\)[/tex] into the formula:
[tex]\[ a_{30} = a_1 + 29d \][/tex]
[tex]\[ a_{30} = -50 + 29 \times 5 \][/tex]
[tex]\[ a_{30} = -50 + 145 \][/tex]
[tex]\[ a_{30} = 95 \][/tex]
4. Determine which term equals 35:
We need to find [tex]\(n\)[/tex] such that:
[tex]\[ a_n = a_1 + (n-1) \times d = 35 \][/tex]
Substitute the known values:
[tex]\[ 35 = -50 + (n-1) \times 5 \][/tex]
[tex]\[ 35 + 50 = (n-1) \times 5 \][/tex]
[tex]\[ 85 = (n-1) \times 5 \][/tex]
[tex]\[ n-1 = \frac{85}{5} \][/tex]
[tex]\[ n-1 = 17 \][/tex]
[tex]\[ n = 18 \][/tex]
### Conclusion:
- The first term ([tex]\(a_1\)[/tex]) is [tex]\(-50\)[/tex].
- The common difference ([tex]\(d\)[/tex]) is [tex]\(5\)[/tex].
- The 30th term ([tex]\(a_{30}\)[/tex]) is [tex]\(95\)[/tex].
- The term that equals [tex]\(35\)[/tex] is the 18th term.
