302 ALGEBRA. Thus the 3d term is a +2d; 6th term is a + 5d; 20th term is a + 19d; and, generally, the pth term is a + (p — 1) d.
If n be the number of terms, and if l denote the last, or nth. term, we have l = a + (n- 1) d.
367. The Sum of n Terms in A. P. Let a denote the first term, d the common difference, and n the number of terms. Also let I denote the last term, and S the required sum; then
S = a + (a + d)--(a + 2f?) + -- + (? - 2 d) + (l - d)+L
and, by writing the series in the reverse order,
S = 1 +{l - d)--(l -2 d) + ->--{-(a -{- 2d) + (a + d)+ a.
Adding together these two series,
2S = (a -i- l)--{a -- l) + {a -- 1)+ '" to n terms = n (a + I), ••• S = l(a + I) (1). Since l = a--(n-l)d (2); .-. S = '^a+(n-l)d . . . (3).
368. In the last article we have three useful formulae (1), (2), (3); in each of these any one of the letters may denote the unknown quantity when the three others are known.
Ex. 1. Find the 20th and 35th terms of the series 38, 36, 34, ....
Here the common difference is 36 — 38, or — 2. .-. the 20th term = 38 + 19 (- 2) = 0; and the 35th term = 38 + 34 ( - 2) = - 30.
Ex. 2. Find the sum of the series 5|, 6|, 8, •.• to 17 terms.
Here the common difference is 1^ ; hence from (3) The sum = ^^{2 x -V- + 10 x li} = V (11 + -0) = ll2^ = 2031.
