Finding the 50th Term of an Arithmetic Progression

In an arithmetic progression (AP), the sum of its first n terms is given by the formula (S_n 12n - n^2). This detailed guide explains how to find the 50th term of this progression, using a step-by-step approach that aligns with best SEO practices.

Step-by-Step Solution

The first step is to find the first term (a) and the common difference (d) of the arithmetic progression, given the sum of the first n terms as Sn.

Step 1: Finding a

To find the first term, we substitute n 1 into the given formula:

S1 12(1) - 12 12 - 1 11

Hence, the first term a 11.

Step 2: Finding d

Next, we find the common difference by substituting n 2 into the formula for the sum of the first two terms:

S2 12(2) - 22 24 - 4 20

The sum of the first two terms can also be expressed as:

S2 a (a d) 2a d

Substituting a 11 into the equation:

20 2(11) d
20 22 d
d 20 - 22 -2

Step 3: General Term of the AP

The general term of an arithmetic progression can be expressed as:

Tn a (n - 1)d

Substituting a 11 and d -2:

Tn 11 (n - 1)(-2)
Tn 11 - 2n 2
Tn 13 - 2n

Step 4: Finding the 50th Term

To find the 50th term, we substitute n 50 into the general term formula:

T50 13 - 2(50) 13 - 100 -87

Therefore, the 50th term of the arithmetic progression is: boxed{-87}.

Alternative Methods

Another approach to find the 50th term is to use the relationship between terms and sums:

Tn Sn - Sn-1

For the 50th term:

S50 12(50) - 502 600 - 2500 -1900
S49 12(49) - 492 588 - 2401 -1813
t50 -1900 - (-1813) -1900 1813 -87

This confirms that the 50th term is -87.

Conclusion

The detailed breakdown of the arithmetic progression and its properties, as well as the mathematical derivations, serve as a comprehensive guide for students and professionals. By understanding the underlying principles, one can solve similar problems efficiently. The SEO-optimized keywords help in improving the visibility and ranking of this content in search engine results.