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Answer :
An AP consists of 50 terms and the \(50^{th}\) term is equal to 106 and \(a_3 = 12\)
Using formula \(a_n = a + (n - 1)d\) , to find nth term of arithmetic progression,
\(\Rightarrow a_{50}\)= a + (50 - 1)d, and
\(a_3\)= a + (3 - 1)d
\(\Rightarrow \) 106 = a + 49d and,
12 = a + 2d
These are equations consisting of two variables.
Using equation 106 = a + 49d,
we get a = 106 - 49d
Putting value of a in the equation 12 = a + 2d,
\(\Rightarrow \) 12 = 106 - 49d + 2d
\(\Rightarrow \) 47d = 94
\(\Rightarrow \) d = 2
Putting value of d in the equation,
a = 106 - 49d,
a = 106 – 49 (2)
= 106 – 98 = 8
Therefore, First term = a = 8 and
Common difference = d = 2
To find \(29^{th}\) term,
we use formula \(a_n = a + (n - 1)d\) which is used to find nth term of arithmetic progression,
\(a_{29}\)
= 8 + (29 - 1) 2
= 8 + 56 = 64
Therefore, 29th term of AP is equal to 64.