Exercise 1. Calculate the number

$(1+11+21+31+41+51+61+71+81+91)+(9+19+29+39+49+59+69+79+89+99)$.

Only a numerical answer is expected here.
## Solution to Exercise 1

The most practical way to proceed is to form pairs, each containing a number ending in 1 and a number ending in 9, because the sum will be a number ending in 0. The best approach is to arrange for the sum of each pair to be the same. Thus, we can write:

$$
\begin{aligned}
& (1+11+21+31+41+51+61+71+81+91)+ \\
& (9+19+29+39+49+59+69+79+89+99) \\
= & \underbrace{(1+99)}_{=100}+(11+89)+(21+79)+(31+69)+(41+59)+ \\
& (51+49)+(61+39)+(71+29)+(81+19)+(91+9) \\
= & 100+100+100+100+100+100+100+100+100+100 \\
= & 1000
\end{aligned}
$$