To find the [tex]\( t_{in} \)[/tex] term from the end in the given arithmetic progression (AP) with first term [tex]\( a = 7 \)[/tex] and common difference [tex]\( d = 3 \)[/tex], and last term [tex]\( l = 184 \)[/tex], we follow these steps:
1. Find the total number of terms in the AP:
The [tex]\( n \)[/tex]th term of an AP is given by:
[tex]\[
a_n = a + (n-1) \cdot d
\][/tex]
Here, [tex]\( a_n = 184 \)[/tex], [tex]\( a = 7 \)[/tex], and [tex]\( d = 3 \)[/tex]. Plugging in these values, we get:
[tex]\[
184 = 7 + (n-1) \cdot 3
\][/tex]
Solve for [tex]\( n \)[/tex]:
[tex]\[
184 - 7 = (n-1) \cdot 3
\][/tex]
[tex]\[
177 = (n-1) \cdot 3
\][/tex]
[tex]\[
n-1 = 59
\][/tex]
[tex]\[
n = 60
\][/tex]
So, there are 60 terms in the AP.
2. Determine the term from the end:
To find the [tex]\( t_{in} \)[/tex] term from the end of the AP, we need to find the [tex]\( (n - t + 1) \)[/tex]th term from the start. For example, if [tex]\( t_{in} = 3 \)[/tex], we need to find the [tex]\( (60 - 3 + 1) \)[/tex]th term from the start.
[tex]\[
\text{term from the start} = 60 - t_{in} + 1
\][/tex]
If [tex]\( t_{in} = 3 \)[/tex]:
[tex]\[
\text{term from the start} = 60 - 3 + 1 = 58
\][/tex]
3. Calculate the term:
The [tex]\( 58 \)[/tex]th term from the start is given by:
[tex]\[
a_{58} = a + (58-1) \cdot d
\][/tex]
Plug in [tex]\( a = 7 \)[/tex] and [tex]\( d = 3 \)[/tex]:
[tex]\[
a_{58} = 7 + 57 \cdot 3
\][/tex]
[tex]\[
a_{58} = 7 + 171 = 178
\][/tex]
Therefore, the [tex]\( t_{in} \)[/tex] term from the end is [tex]\( 178 \)[/tex].
