Arithmetic Series - Tn

Appreciate: The n^th term of an arithmetic sequence/series equals the (n-1)^th term + difference (that is T_n = T_n-1 + d).

Remember: When you are given a particular term, always (nearly anyway) write it out in symbols.

Example: if you are given the 5^th term is 20 - then you write T₅ = T₁ + 4d = 20 and go on from there.

Also: if you are told a sequence/series is "in arithmetic progression", that can only mean there is a common difference between the terms you are given.

So find the common difference by T₂ - T₁ = T₃ - T₂ etc.

Finding a term	1. We are given the relationship that T_n = 33 + 6n. Find T₂₂. Answer.22nd term is 165.	2. Find the 40^th term in the sequence 5, 8, 11, ... Hint.Write down T₁ and d and say the n^th term to yourself. Answer.40th term is 122.
	3. An arithmetic sequence has -3 and 5 as two consecutive terms. The 18^th term is 101. Write down the 17^th and 20^th terms. Hint.These 2 terms are not necessarily the 1st and 2nd terms. But they are consecutive so we can find the common difference. Answer.17th term is 93 and 20th term is 117.	4. The 1^st term in an arithmetic sequence is 21 and the common difference is -10. Find the 17^th term. Answer.17th term is -139.
Finding a 1^st term	5. An arithmetic sequence has a common difference of 4 and its 27^th term is 117. Find the 1^st term. Answer.1st term is 13.	6. An arithmetic sequence has a common difference of -17 and its 23^th term is -352. Find the 1^st term. Answer.1st term is 22.
Finding a difference	7. The first term in an arithmetic sequence is 4 and the 25^th term is 172. Find the common difference. Answer.Common difference is 7.	8. The first term in an arithmetic sequence is 42 and the 18^th term is 33.5. Find the common difference. Answer.The common difference is -0.5.
Finding the number of terms	9. Find the number of terms in the arithmetic sequence 10, 7, 4, ..., -47. Answer.n = 20.	10. Which term is -372 if it is part of an arithmetic series starting at 240 and having a common difference of -9? Answer.Term 69.
	11. Find the number of multiples of 7 between 300 and 650. Hint.You need the first multiple of 7. So divide 300 by 7 = 42.9 then round up to 43. 1st multiple is 43×7 = 301. Repeat for last multiple and round down (650/7 = 92.9) and 92×7 = 644 These are the 1st and last terms - find n. Answer.n = 50.	12. What is the first term in the sequence -8, 5, 18, ... greater than 1250? Answer.Term 98 is 1253.
Given 2 terms	13. If the 6^th term of an arithmetic sequence is 17 and the 13^th term is 80, what are the first three terms in the sequence? Answer.-28, -19, -10.	14. If the 4^th term of an arithmetic sequence is 27 and the 7^th term is 12, what is the common difference for the sequence? Answer.Difference is -5.
	15. For a particular arithmetic sequence, three times the 3rd term equals the 9th term. What could be the first three terms of a possible sequence with these properties? Answer.The terms could be 2, 4 and 6.	16. If the 2^nd term of an arithmetic sequence is 7 and the 7^th term is 52, find the value of the first term greater than 1,000. Answer.113th term is 1006.
Miscellaneous	17. The terms x-2, 2x-1, 4x-4 form an arithmetic sequence. Find the first three terms of the sequence. Hint.Find the common difference and equate the two expressions. Answer.First 3 terms are 2, 7 and 12.	18. Find the number of multiples of 13 between 350 and 730. Answer.There are 30 multiples.
	19. The sum of the 2^nd and 5^th terms of an arithmetic series is 32 while the sum of the 3^rd and 8^th terms is 48. (i) find the 1^st term and the common difference. (ii) find the 10th term. Hint.Just write expressions for the 2^nd and 5^th terms, add them and put the result = 32. Do the same for the other two terms and solve simultaneously. Answer.The 10th term is 42.	20. In an arithmetic sequence, the 10^th term is double the 6^th term. The square of the 3^rd term is equal to the 6^th term. Find the first three terms in the sequence. Answer.The first three terms are -4, 0 and 4.