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ARITHMETICAL PROGRESSION. 305 372. Between two given quantities it is always possible to insert any number of terms such that the whole series thus formed shall be in A. P.; and by an extension of the definition in Art. 370, the terms thus inserted are called the arithmetic means.

Ex. Insert 20 arithmetic means between 4 and 67.

Including the extremes the number of terms will be 22; so that we have to find a series of 22 terms in A. P., of which 4 is the first and 67 the last.

Let d be the common difference;

then 67 = the 22d term, = 4 + 21d;

whence d = 3, and the series is 4, 7, 10,... 61, 64, 67; and the required means are 7, 10, 13,... 58, 61, 64. 373. To insert a given number of arithmetic means between two given quantities.

Let a and b be the given quantities, m the number of means.

Including the extremes the number of terms will be m + 2; so that we have to find a series of m + 2 terms in A. P., of which a is the first, and b is the last.

Let d be the common difference;

then b = the (m + 2)th term = a + (m - 1) d; whence d = b - a m + 1;

and the required means are a + (b - a) (m + 1), a + 2 b - a m + 1 ,..., a + m b - a m + 1 Ex. 1. Find the 30th term of an A. P. of which the first term is 17, and the 100th term -16.

Let d be the common difference; then -16 = the 100th term = 17 + 99 d; d= -1 3. The 30th term = 17 + 29(- 1 3) = 7 1 3.