Next we add up the terms of an arithmetic sequence. The scratch work below is where the formula for the sum comes from.
Remark (Scratch work for the next problem). Summing (common difference ). First tries: pairing, and ; then with :
Fixing the formula: the th term is , so the sum is
which gives
Pairing again, and ;?
Problem 11.2.1..
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,,, so
By the formula :
Problem 11.2.2. The page numbers of a book are added up, with one page number counted twice, and the total is . Which page was counted twice?
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therefore page .
This is the only one, since , and , which cannot be, and , which cannot be either.
Problem 11.2.3. Find all such that .
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. With terms, , so the sum is
Since , try and :.
: the 17th term is .
: the 30th term is .
Problem 11.2.4. Let be in AP with , and let
If , what is ?
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The numerator is , all over , giving . So:
If , then
Example 11.2.5. The sum of the first terms of an AP is and the sum of the first terms is , where for all . Find the ratio of the 8th term to the 23rd term.
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WLOG let (for simplicity). The sums:
terms
sum
The sum of the first terms:
so for :, and
In general,
so , or .
Then
Exercises
Exercise 11.2.6.
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So there are terms, and the sum is
Exercise 11.2.7.Show solution
Exercise 11.2.8. Let . Find the sum of the smallest odd positive integers, and calculate it for .
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Say :
For :
Exercise 11.2.9. Let and be positive integers with . Find the sum of the positive integers from to , inclusive, in terms of and .
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If only : since is the th term, is the 2nd term, so and
In general,
Remark (Correction). This only recovers the two-term case. For the integers from to the common difference is , not , and there are terms, so the sum is
Exercise 11.2.10. For all , the sum of the first terms of a certain AP is . Find a formula for the th term.