What is the General Term of an Arithmetic Sequence?

Let's explore the problem of finding the general term $T_n$ for an arithmetic sequence given certain conditions. This can be a challenging but rewarding task, especially when we have specific values of terms in the sequence. We'll delve into the details and solve a problem step-by-step.

Problem Statement

Given an arithmetic sequence where $t_5 -2$ and $T_{12} 12 frac{1}{2}$, we need to find the general term $T_n$.

Step-by-Step Solution

First, let's clarify the given information:

$t_5 a 4d -2$ $T_{12} 12 frac{1}{2} frac{25}{2}$

Recall that the sum of the first 12 terms $S_{12}$ can be represented as:

$S_{12} frac{12}{2} cdot (2a 11d) 25/2$

Which simplifies to:

$6(2a 11d) frac{25}{2}$

Simplifying further:

$2(2a 11d) frac{25}{12}$

Therefore:

$4a 22d frac{25}{12}$

Using the Given Information

We now have two equations:

$a 4d -2$ $4a 22d frac{25}{12}$

Let's solve these equations simultaneously.

Elimination Method

First, multiply the first equation by 4:

$4a 16d -8$

Now, we have:

$4a 16d -8$ $4a 22d frac{25}{12}$

Subtracting the first equation from the second equation:

$6d frac{25}{12} 8$

$6d frac{25}{12} frac{96}{12}$

$6d frac{121}{12}$

$d frac{121}{12} cdot frac{1}{6}$

$d frac{121}{72} -frac{23}{36}$

Substituting the Value of d

Now, substitute $d -frac{23}{36}$ into the first equation:

$a 4left(-frac{23}{36}right) -2$

$a - frac{92}{36} -2$

$a - frac{23}{9} -2$

$a -2 frac{23}{9} frac{-18 23}{9} frac{5}{9}$

General Term of the Sequence

The general term $T_n$ of an arithmetic sequence is given by:

$T_n a (n-1)d$

Substitute $a frac{5}{9}$ and $d -frac{23}{36}$ into the formula:

$T_n frac{5}{9} (n-1)left(-frac{23}{36}right)$

$T_n frac{5}{9} - frac{23(n-1)}{36}$

This simplifies to:

$T_n frac{20 - 23(n-1)}{36} frac{43 - 23n}{36}$

Conclusion

We have successfully found the general term of the given arithmetic sequence:

$T_n frac{43 - 23n}{36}$

This method can be applied to other similar problems, providing a solid foundation for understanding arithmetic sequences and their properties.

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