Answer :
Let's solve the problem step-by-step where the third and fifth terms of an arithmetic progression (AP) are given as 7 and 13, respectively. We need to find the AP.
In an AP, each term after the first is the previous term plus a constant difference called the common difference, denoted by [tex]\( d \)[/tex]. The formula for the [tex]\( n \)[/tex]-th term of an AP is:
[tex]\[ T_n = a + (n - 1) \cdot d \][/tex]
where:
- [tex]\( T_n \)[/tex] is the [tex]\( n \)[/tex]-th term,
- [tex]\( a \)[/tex] is the first term,
- [tex]\( d \)[/tex] is the common difference.
Let's label the equations given by the problem:
1. Third term: [tex]\( T_3 = a + 2d = 7 \)[/tex]
2. Fifth term: [tex]\( T_5 = a + 4d = 13 \)[/tex]
Now, we'll solve these equations to find [tex]\( a \)[/tex] (the first term) and [tex]\( d \)[/tex] (the common difference).
Step 1: Let's eliminate [tex]\( a \)[/tex] by subtracting the equation for the third term from the equation for the fifth term. This eliminates [tex]\( a \)[/tex] and helps us find [tex]\( d \)[/tex] directly:
[tex]\[ (a + 4d) - (a + 2d) = 13 - 7 \][/tex]
[tex]\[ 2d = 6 \][/tex]
Step 2: Solve for [tex]\( d \)[/tex]:
[tex]\[ d = \frac{6}{2} = 3 \][/tex]
Step 3: Substitute [tex]\( d = 3 \)[/tex] back into the equation for the third term to find [tex]\( a \)[/tex]:
[tex]\[ a + 2d = 7 \][/tex]
[tex]\[ a + 2 \cdot 3 = 7 \][/tex]
[tex]\[ a + 6 = 7 \][/tex]
[tex]\[ a = 7 - 6 = 1 \][/tex]
So, the first term of the arithmetic progression is 1, and the common difference is 3.
Thus, the arithmetic progression starts with 1 and adds 3 to each term to get the next term.
In an AP, each term after the first is the previous term plus a constant difference called the common difference, denoted by [tex]\( d \)[/tex]. The formula for the [tex]\( n \)[/tex]-th term of an AP is:
[tex]\[ T_n = a + (n - 1) \cdot d \][/tex]
where:
- [tex]\( T_n \)[/tex] is the [tex]\( n \)[/tex]-th term,
- [tex]\( a \)[/tex] is the first term,
- [tex]\( d \)[/tex] is the common difference.
Let's label the equations given by the problem:
1. Third term: [tex]\( T_3 = a + 2d = 7 \)[/tex]
2. Fifth term: [tex]\( T_5 = a + 4d = 13 \)[/tex]
Now, we'll solve these equations to find [tex]\( a \)[/tex] (the first term) and [tex]\( d \)[/tex] (the common difference).
Step 1: Let's eliminate [tex]\( a \)[/tex] by subtracting the equation for the third term from the equation for the fifth term. This eliminates [tex]\( a \)[/tex] and helps us find [tex]\( d \)[/tex] directly:
[tex]\[ (a + 4d) - (a + 2d) = 13 - 7 \][/tex]
[tex]\[ 2d = 6 \][/tex]
Step 2: Solve for [tex]\( d \)[/tex]:
[tex]\[ d = \frac{6}{2} = 3 \][/tex]
Step 3: Substitute [tex]\( d = 3 \)[/tex] back into the equation for the third term to find [tex]\( a \)[/tex]:
[tex]\[ a + 2d = 7 \][/tex]
[tex]\[ a + 2 \cdot 3 = 7 \][/tex]
[tex]\[ a + 6 = 7 \][/tex]
[tex]\[ a = 7 - 6 = 1 \][/tex]
So, the first term of the arithmetic progression is 1, and the common difference is 3.
Thus, the arithmetic progression starts with 1 and adds 3 to each term to get the next term.
