To find the 15th term of an Arithmetic Progression (AP) where the first term is given as [tex]\( a = 42 \)[/tex] and the 10th term ([tex]\( T_{10} \)[/tex]) is 210, we can follow these steps:
1. Determine the common difference [tex]\( d \)[/tex]:
The formula for the nth term [tex]\( T_n \)[/tex] of an AP is given by:
[tex]\[
T_n = a + (n - 1) \cdot d
\][/tex]
For the 10th term:
[tex]\[
T_{10} = a + 9d
\][/tex]
Substituting the given values:
[tex]\[
210 = 42 + 9d
\][/tex]
Solving for [tex]\( d \)[/tex]:
[tex]\[
210 - 42 = 9d
\][/tex]
[tex]\[
168 = 9d
\][/tex]
[tex]\[
d = \frac{168}{9}
\][/tex]
Thus, the common difference [tex]\( d \)[/tex] is approximately [tex]\( 18.666666666666668 \)[/tex].
2. Calculate the 15th term ([tex]\( T_{15} \)[/tex]):
Using the formula [tex]\( T_n = a + (n - 1) \cdot d \)[/tex] again:
[tex]\[
T_{15} = a + 14d
\][/tex]
Substituting the values for [tex]\( a \)[/tex] and [tex]\( d \)[/tex]:
[tex]\[
T_{15} = 42 + 14 \cdot 18.666666666666668
\][/tex]
Evaluating the expression:
[tex]\[
T_{15} = 42 + 261.33333333333337
\][/tex]
[tex]\[
T_{15} = 303.33333333333337
\][/tex]
Therefore, the 15th term of the Arithmetic Progression is [tex]\( 303.33333333333337 \)[/tex].
