下面给出了一个数列的前四项。

1. Continue the given pattern to form and evaluate $T_5$.

From the pattern,
$T_5=\dfrac{6}{5^4}-\dfrac{7}{5^5}$
$=\dfrac{30}{5^5}-\dfrac{7}{5^5}$
$=\dfrac{23}{5^5}=\dfrac{23}{3125}$

1. Identify the general form of the two fractions in $T_n$.

From the pattern,
$T_1=\frac{2}{1}-\frac{3}{5},\quad T_2=\frac{3}{5}-\frac{4}{5^2},\quad T_3=\frac{4}{5^2}-\frac{5}{5^3}.$
So,
$T_n=\frac{n+1}{5^{n-1}}-\frac{n+2}{5^n}.$

2. Simplify the expression into a single fraction.

$T_n=\frac{n+1}{5^{n-1}}-\frac{n+2}{5^n}$
$=\frac{5(n+1)}{5^n}-\frac{n+2}{5^n}$
$=\frac{5n+5-n-2}{5^n}$
$=\frac{4n+3}{5^n}.$

1. Write the first ten terms using the subtraction pattern.

From the given pattern,
$T_n=\frac{n+1}{5^{n-1}}-\frac{n+2}{5^n}.$
So

2. Cancel matching terms and evaluate the remaining expression.

All the middle terms cancel:
$T_1+T_2+\cdots+T_{10}=2-\frac{12}{5^{10}}.$