Answer :
To solve this question, let's break it down into two parts:
Part 1: Find the Term that is 4 and -8 in the AP (Arithmetic Progression)
Given that the third term [tex]a_3[/tex] of an arithmetic progression (AP) is 4 and the ninth term [tex]a_9[/tex] is -8, we need to determine which term of this AP is 4.
Express the Terms in Terms of [tex]a[/tex] and [tex]d[/tex]:
- The general form of the [tex]n[/tex]-th term of an AP is given by:
[tex]a_n = a + (n-1)d[/tex] - For the third term:
[tex]a + 2d = 4[/tex] - For the ninth term:
[tex]a + 8d = -8[/tex]
- The general form of the [tex]n[/tex]-th term of an AP is given by:
Solve the System of Equations:
- We have the equations:
[tex]a + 2d = 4[/tex]
[tex]a + 8d = -8[/tex] - Subtract the first equation from the second:
[tex](a + 8d) - (a + 2d) = -8 - 4[/tex]
[tex]6d = -12[/tex]
[tex]d = -2[/tex] - Substitute [tex]d = -2[/tex] back into the first equation:
[tex]a + 2(-2) = 4[/tex]
[tex]a - 4 = 4[/tex]
[tex]a = 8[/tex]
- We have the equations:
Find the Term Number Where the Term is 4:
- Set [tex]a_n = 4[/tex]:
[tex]a + (n-1)d = 4[/tex]
[tex]8 + (n-1)(-2) = 4[/tex]
[tex]8 - 2(n-1) = 4[/tex]
[tex]8 - 2n + 2 = 4[/tex]
[tex]10 - 2n = 4[/tex]
[tex]-2n = -6[/tex]
[tex]n = 3[/tex] - Therefore, the third term is indeed 4 as confirmed earlier.
- Set [tex]a_n = 4[/tex]:
Part 2: Find the 8th Term of the Given AP
For the arithmetic progression given by the sequence [tex]7, 10, 13, \ldots[/tex], we want to find the 8th term.
Identify [tex]a[/tex] and [tex]d[/tex] in the AP:
- Here, [tex]a = 7[/tex] and [tex]d = 10 - 7 = 3[/tex].
Use the General Term Formula:
- The formula for the [tex]n[/tex]-th term is:
[tex]a_n = a + (n-1)d[/tex] - For the 8th term:
[tex]a_8 = 7 + (8-1) \times 3[/tex]
[tex]a_8 = 7 + 21[/tex]
[tex]a_8 = 28[/tex] - Thus, the 8th term in this progression is 28.
- The formula for the [tex]n[/tex]-th term is:
Therefore, the solution to the given problem is that the third term of the AP is the term which is 4, and the 8th term of the sequence 7, 10, 13, ... is 28.
