An AP consists of 50 terms in which the term is 12 and the last term is 106. Find the term. Hint: If 'a' is the first term and 'd' the common difference, then we arrive at the equations and . Solve this pair of linear equations for 'a' and 'd'.
64
Explanation
To find the 29th term of the AP, we first need to determine the first term 'a' and the common difference 'd'. Using the given information about the 3rd term and the last (50th) term, we can set up a pair of linear equations as suggested by the hint. Solving these equations gives us the values of 'a' and 'd', which can then be substituted into the general formula for the nth term of an AP, t_n = a + (n - 1)d, to find the 29th term.
Solution Steps
Given equations: a + 2d = 12 and a + 49d = 106
Subtracting the first equation from the second: (a + 49d) - (a + 2d) = 106 - 12
Solving for d: 47d = 94, which gives d = 2
Substituting d = 2 into the first equation: a + 2(2) = 12, so a + 4 = 12, which gives a = 8
Finding the 29th term using t_n = a + (n - 1)d: t_29 = 8 + (29 - 1) × 2
Calculating the final value: t_29 = 8 + = 8 + 56 = 64