XIII.

A - a = b - A, lintwhence A = a + b2. rintFormula (2)

Hence, the arithmetical mean of any two numbers is found by taking half their sum.

Sometimes it is required to insert several arithmetical means between two numbers.

If $m=$the number of means, and $n=$the whole number of terms, then $m+2=n$. If $m+2$ is substituted for $n$ in formula (1),

l = a + (n - 1)d, lintthe result is l = a + (m + 1)d. [1] lint By transposing $a$, l - a = (m + 1) d. l - am + 1 = d. Formula (3)

Thus, if it be required to insert six means between $3$ and $17$, the value of $d$ is found to be ${\displaystyle \frac{17-3}{6+1}}=2$; and the series will be $3$, $5$, $7$, $9$, $11$, $13$, $15$, $17$.

Exercise 75.

1. Find the 25th term in the series $3$, $6$, $9$, $\dots$.

2. Find the 13th term in the series $50$, $49$, $48$, $\dots$.

3. Find the 15th term in the series $\frac{1}{7}$, $\frac{3}{7}$, $\frac{5}{7}$, $\dots$.

4. Find the 19th term in the series $\frac{1}{4}$, $-\frac{1}{4}$, $-\frac{3}{4}$, $\dots$.

5. Find the 10th term in an arithmetical progression whose 1st term is $5$ and 3d term $9$.

6. Find the 11th term in an arithmetical progression whose 1st term is $10$ and whose 6th term is $5$.

7. If the 3d term of an arithmetical progression is $20$ and the 13th term is $100$, what is the 20th term?

8. Which term of the series $5$, $7$, $9$, $11$, $\dots$, is $43$?

9. Which term of the series $\frac{4}{3}$, $\frac{3}{2}$, $\frac{5}{3}$, $\dots$, is $18$?

10. What is the arithmetical mean of $20$ and $32$?

11. What is the arithmetical mean of $a+b$ and $a-b$?

12. Insert $8$ arithmetical means between $20$ and $29$.

171. To Find the Sum of Any Number of Terms of an Arithmetical Series.

If $l$ denote the last term, $a$ the first term, $n$ the number of terms, $d$ the common difference, and $s$ the sum of the terms, it is evident that the series beginning with the first term will be $a$, $a+d$, $a+2d$, etc., and beginning with the last term will be $l$, $l-d$, $l-2d$, etc. Therefore,

{r*{12}{c}} s &=& a &+& (a + d) &+& (a + 2d) &+& \dots &+& (l - d) &+& l, \rlap{\quad\text{or}} \ s &=& l &+& (l - d) &+& (l - 2d) &+& \dots &+& (a + d) &+& a \ \hline 2s &=& (a + l) &+& (a + l) &+&