An AP consists of 50 terms of which 3rd term is 12 and the last term is 106. Find the 29thterm

To find the 29th term of an arithmetic progression (AP) with 50 terms, given that the 3rd term is 12 and the last term is 106, we can use the formula for the nth term of an AP:

nth term = a + (n - 1)d

where a is the first term, n is the term number, and d is the common difference.

First, let's find the common difference (d). We know that the 3rd term is 12, so we can substitute these values into the formula:

12 = a + (3 - 1)d

Simplifying this equation, we get:

12 = a + 2d

Next, we can find the first term (a) using the formula for the last term of an AP:

last term = a + (n - 1)d

Substituting the given values, we have:

106 = a + (50 - 1)d

Simplifying this equation, we get:

106 = a + 49d

Now we have two equations:

12 = a + 2d 106 = a + 49d

We can solve these equations simultaneously to find the values of a and d.

Subtracting the first equation from the second equation, we get:

106 - 12 = (a + 49d) - (a + 2d)

Simplifying this equation, we get:

94 = 47d

Dividing both sides by 47, we find:

d = 2

Now we can substitute the value of d back into the first equation to find the value of a:

12 = a + 2(2)

Simplifying this equation, we get:

12 = a + 4

Subtracting 4 from both sides, we find:

a = 8

Now that we have the values of a and d, we can find the 29th term using the formula:

29th term = a + (29 - 1)d

Substituting the values, we have:

29th term = 8 + (29 - 1)(2)

Simplifying this equation, we get:

29th term = 8 + 28(2)

Calculating this, we find:

29th term = 8 + 56

Therefore, the 29th term of the given arithmetic progression is 64.