Given Information:
- An AP (Arithmetic Progression) consists of 50 terms.
- The 3rd term of the AP is 12.
- The last term of the AP is 106.

Objective:
To find the 29th term of the AP.




Approach:
To find the 29th term of the AP, we need to calculate the common difference (d) and then use the formula for the nth term of an AP.




Finding the Common Difference:
To find the common difference (d), we can use the formula:

d = (last term - first term) / (number of terms - 1)

Using the given information, we can substitute the values:

d = (106 - first term) / (50 - 1)




Finding the First Term:
We are given the 3rd term of the AP, which is 12. To find the first term, we can use the formula:

first term = 3rd term - 2 * d

Substituting the values:

12 = 3 - 2 * d
12 = 3 - 2 * ((106 - first term) / (50 - 1))

Simplifying the equation:

12 = 3 - 2 * (106 - first term) / 49
12 * 49 = 3 * 49 - 2 * (106 - first term)
588 = 147 - 2 * (106 - first term)
588 = 147 - 212 + 2 * first term
588 = -65 + 2 * first term
2 * first term = 588 + 65
2 * first term = 653
first term = 653 / 2
first term = 326.5

Since the AP consists of integer terms, we can round down the first term to the nearest whole number:

first term = 326




Calculating the Common Difference:
Now that we know the first term, we can substitute the values into the common difference formula:

d = (106 - first term) / (50 - 1)
d = (106 - 326) / 49
d = -220 / 49
d ≈ -4.4898

Since the common difference is approximately -4.4898, we can round it to the nearest whole number:

d = -4




Finding the 29th Term:
Now that we know the first term (326) and the common difference (-4), we can use the formula for the nth term of an AP:

nth term = first term + (n - 1) * d

Substituting the values:

29th term = 326 + (29 - 1) * (-4)
29th term = 326 + 28 * (-4)
29th term = 326 - 112
29th term = 214

Therefore, the 29th term of the AP is 214.