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To find the last Term.

CHAPTER VII.

PROGRESSIONS.

SECTION I.

Arithmetical Progression.

244. An Arithmetical Progression, or a progression by differences, is a series of terms or quantities which continually increase or decrease by a constant quantity.

This constant increment or decrement is called the common difference of the progression.

Throughout this section the following notation will be retained. We shall use

a = the first term of the progression,

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245. Problem. To find the last term of an arithmetical progression when its first term, common difference, and number of terms are known.

Solution. In this case a, r, and n, are supposed to be known, and 7 is to be found. Now the successive terms of the series if it is increasing are

a, a +r, a +2 r, a+3r, a +4r, &c.;

Sum of two Terms equally distant from the extremes.

so that the nth term is obviously

l= a + (n − 1) r.

But if the series is decreasing, the last term must be

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Both these cases are, however, included in one, if we suppose r to be negative when the series is decreasing.

246. Corollary. In like manner any other term,

such as

the mth, is

a (m1) r.
+

247. Corollary. By writing the series in an inverted order, beginning with the last term, a new series is found, of which the first term is 7, and the common difference -r. Hence the mth term of this series, that is, the mth term counting from the last of the given series, is

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248. Corollary. The sum of the mth term and of the mth term from the last is, therefore,

[a + (m − 1) r] + [l — (m − 1) r] = a + 1;

that is, the sum of any two terms, taken at equal distances from the two extremes of an arithmetical series, is equal to the sum of the two extremes.

249. Problem. To find the sum of an arithmetical progression when its first term, last term, and number of terms are known.

Solution. In this case, a, l, and n are supposed to be known, and S is to be found.

To find the Sum of the Progression.

Suppose the terms of the series to be written as follows, first in the regular order, and then in an inverted order:

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The sum of the terms of each of these progressions being S, the sum of both of them must be 2 S, that is, 28=(a+1)+(b+k)+(c+i)...+(i+c)+(k+b)+(l+a).

But by the preceding corollary, we have

a+l=b+k=c+i=&c.

Hence 2 S is equal to as many times (a+1) as there are terms in the series, that is,

28= (a+1)n;

or S=(a+1)n;

that is, the sum of a progression is equal to half the sum of the two extremes, multiplied by the number of terms.

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either two of the quantities a, l, r, n, and S can be determined when the other three are known.

251. EXAMPLES.

1. Find the 100th term of the series 2, 9, 16, &c.

2. Find the sum of the preceding series.

Ans. 695.

3. Find S, when a, r, and n are known.

Ans. 34850.

Ans. S[2a+(n−1)r] n.

Examples in Progression.

4. Find n and S, when a, 1, and r are known.

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5. Find the number and sum of terms of the series of which the first term is 6, the last term 796, and the common difference 10.

Ans. The number of terms = 80, the sum 32080.

6. Find r, when a, 7, and n are known.

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7. Find the common difference and sum of the series, of which the first term is 75, and the last term 15, and the number of terms 13.

Ans. The common difference —— 5,

the sum = 585.

8. Find r and n, when a, l, and S are known.

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9. Find the common difference and number of terms of a series, of which the first term is 2, and the last term 345, and the sum 8675.

Ans. The number of terms = 50, the common difference 7.

10. Find and n, when a, r, and S are known.

Ans. n =

√ [2r8+(a—}r)2] — (a — § r) ̧

r

l=√[2rS+(a — { r)2] — } r.

Examples in Progression.

11. Find the last term and number of terms of a series, of which the first term is 3, the common difference 4, and the sum of the terms 105.

Ans. The last term = 27,

the number of terms = 7.

12. Find a and n, when 1, r, and S are known.

Ans. n =

?+}r=v[(1+}r)2 —2r8]

a=±v[(1+{r)2−2r8]+{r.

13. Find the first term and the number of terms of a series, of which the last term is 13, the common difference 3, and the sum of the series 35.

Ans. The first term = 1,

the number of terms = 5.

14. Find / and r, when a, n, and S are known.

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15. Find the last term and common difference of a series, of which the first term is, the number of terms 12, and the sum 100.

Ans. The last term = 16,

the common difference = 148.

16. Find a and r, when 7, n, and S are known.

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