XIII.

(a + l) &+& \dots &+& (a + l) &+& (a + l) \ 2s &=& \multicolumn{11}{l}{\text{ ( a + l ) taken as many times as there are \emph{terms},}} \displaybreak[1] \ [b] 2s &= n(a + l), \ \text{and } s &= \frac{n}{2}(a + l).

\tag*{Formula (4)}

Putting for $l$ its value $a+(n-1)d$, in formula (4), we have

$$\begin{gathered} s\&=\frac{n}{2}\backslash \{a+a+(n-1)d\backslash \} \\ \&=\frac{n}{2}\backslash \{2a+(n-1)d\backslash \}\text{Formula (5)} \end{gathered}$$

1. Find the sum of the first $16$ terms of the series $5$, $7$, $9$, $11$.

lint Here a = 5, d = 2, n = 16. Putting these values in formula (5) we have s = 162(10 + 15 × 2) = 320

2. Show that the sum of any number of odd numbers, beginning with $1$, is a square number.

The series of odd numbers is $1$, $3$, $5$, $7$, $\dots$.

lint Here a = 1 d = 2. Putting these values in formula (5) we have s = n2 \2 + (n - 1)2\ = n2 × 2n = n^2.

Therefore the sum of the first $5$ odd numbers is $5^2$ or $25$, of the first $8$ odd numbers is $8^2$ or $64$, and so on.

3. The sum of $20$ terms of an arithmetical progression is $420$, and the first term is $2$. Find the common difference.

lint Here s = 420, n = 20, a = 2. Putting these values in formula (5), we have 420 = 202(4 + 19d) = 40 + 190d 190d = 380 d = 2.

Therefore the common difference is $2$.

Exercise 76.

1. Find the sum of $3$, $5$, $7$, $\dots$, to $20$ terms.

2. Find the sum of $14$, $14\frac{1}{2}$, $15$, $\dots$, to $12$ terms.

3. Find the sum of $\frac{7}{6}$, $1$, $\frac{5}{6}$, $\dots$, to $10$ terms.

4. Find the sum of $-7$, $-5$, $-3$, $\dots$, to $16$ terms.

5. Find the sum of $12$, $9$, $6$, $\dots$, to $21$ terms.

6. Find the sum of $-10\frac{1}{2}$, $-9$, $-7\frac{1}{2}$, $\dots$, to $25$ terms.

7. The sum of three numbers in arithmetical progression is $9$, and the sum of their squares is $35$. Find the numbers.

Let $x-y$, $x$, $x+y$, stand for the numbers.

8. A common clock strikes the hours from $1$ to $12$. How many times does it strike every $24$ hours?