Solving Arithmetic Progressions with Given Sum and Common Difference
An arithmetic progression (AP) is a sequence of numbers such that the difference between any two successive members is constant. Given the sum of the first 9 terms of an AP is 72 and the common difference (d) is 5, let's solve for the first term ((t_1)) and the 10th term ((t_{10})).

Sum of the First 9 Terms

We are given that the sum of the first 9 terms ((S_9)) is 72. The formula for the sum of the first (n) terms of an AP is:

[S_n frac{n}{2} times (2a (n - 1)d)]

Substituting the given values:

[S_9 frac{9}{2} times (2a (9 - 1) times 5) 72]

Simplify the equation:

[72 frac{9}{2} times (2a 40)]

Multiplying both sides by 2 to clear the fraction:

[144 9 times (2a 40)]

Divide both sides by 9:

[16 2a 40]

Solve for (a):

[2a 16 - 40] [a -12]

So, the first term ((t_1)) is (-12).

Finding the 10th Term

Now, we need to find the 10th term ((t_{10})). The (n)th term of an AP can be found using the formula:

[t_n a (n - 1)d]

For the 10th term, (n 10):

[t_{10} -12 (10 - 1) times 5]

Simplifying this, we get:

[t_{10} -12 9 times 5] [t_{10} -12 45] [t_{10} 33]

Thus, the first term is (-12) and the 10th term is (33).

General Term and Summation Formulas

The general term of the sequence is given by:

[t_n -12 (n - 1)5]

Or, simplifying further:

[t_n -12 5n - 5] [t_n 5n - 17]

The sum of the first (n) terms of the sequence can be expressed as:

[S_n frac{n}{2} times (2 times -12 (n - 1) times 5)]

Or, simplifying:

[S_n frac{n}{2} times (5n - 29)]

Or, in an alternative form:

[S_n frac{5n^2 - 29n}{2}]

The terms and formulas provide a comprehensive understanding of arithmetic progressions using the given sum and common difference.

Keywords: arithmetic progression, common difference, sum of terms