Series - Arith - Summation

The focus of the questions on this page is:

last term is known;
last term unknown;
sum of multiples;
using the sum to find a given term or the difference;
using the sum to find the number of terms.

Direct sum of n terms (last term known).	1. The first term of an arithmetic series is 4 and the 12th term is 40. What is the sum of the first 12 terms? AnswerSum = 264..	2. Write the sum of the series log_e2 + log_e4 + log_e8 + ... + log_e 512 in the form a log_eb where a and b are integers greater than 1. AnswerSum = 45log_e 2..
	3. Find the sum of 12 terms of the following logarithmic series: ln(x) + ln(x²) + ln(x³) + .. Answer.Sum = 78 ln x.	4. Weights of equal mass are being placed in a floating container. First 3 weights are placed in the container and then another 5 and then another 7 and so on. Finally when the stage comes when another 43 weights are placed in the container, the container sinks. How many weights will be in the container when it sinks? Answer.After 20 additions there will be 460 weights.
Direct sum of n terms (last term unknown).	5. Find the sum of the first 51 positive odd integers. Answer. Sum of 51 numbers = 2,601.	6. Ellen will be starting Year 12 in a few months and she wants to save money to give her some extra spending power for her holiday in northern NSW after the HSC. She places $20 into a jar at the start of July and makes additional contributions each month for 18 months. Her additional contributions increase in value by $2 per month. How much has Ellen saved at the end of the 18 months? Answer.Ellen had $666 in the jar!!.
	7. A student has a two week break. She decides to watch 15 minutes of Episode 1 of a Netflix thriller on the first day of her break. The next day she watches 20 minutes and increases her watching time by 5 minutes each day (she has great self-control!!). The thriller has five series with eight 25 minute episodes in each. The break gives her 16 days (including weekends) to watch. (i) Does she finish all five series during her break (without binging at the end)? If not finished, approximately where about is she at? (ii) If she is not finished during the break, what daily increase in viewing time should she have planned so as to complete all 40 episodes in the 16 days? Answer.(i) No - she has viewed 840 minutes and so is just over half way through Episode 2 of Series 5. (ii) Increase from 5 minutes to 6 mins 20 secs per day.
	8. During an examination, the supervisor stands 3 metres in front of the first desk in a column of desks, holding a box of tissues. Students are seated in columns of desks with 1 metre from the front of each desk to the front of the next desk. There are 25 desks in a column. All 25 students in a particular column of desks take turns asking for a tissue. Each student waits until the supervisor returns to the supervising position 3 metres in front of the first desk in the column before putting up their hand. How many metres will the supervisor have to walk to provide each of the students with one tissue and return to the position 3 metres in front of the first desk? Answer.Distance is 648 m.
	9. The first three terms of an arithmetic series are 6 + 10 + 14. Calculate the number of terms required for the sum to exceed 390. Answer. Need 14 terms.	10. Find the sum of 10 terms of the series log_m3 + log_m6 + log_m12 + ... given that log_m2 = 0.43 and log_m3 = 0.68. Answer. Sum = 26.15.
Sum of multiples	11. (i) Find the first and last multiples of 7 between 250 and 370. (ii) Find how many multiples of 7 there are between 250 and 370. (iii) Find the sum of all the multiples of 7 between 250 and 370. Answer.(i) 17 multiples of 7. (ii) Sum of multiples = 5,236.	12. Find the sum of the multiples of 9 between 300 and 650. Answer.There are 39 multiples of 9 and their sum is 18,603.
Using the sum to find terms or difference.	13. If the 5th term of an arithmetic series is 14 and the sum of the first 10 terms is 165, find the first term and the common difference. Answer.1st term = -6 common difference = 5.	14. Show that, if the sum of n terms of the series 15 + 13 + 11 +...is 55, then it may have two possible values for n. Explain why this is so. Answer.The two values for the number of terms are 5 and 11.
	15. The sum of 16 terms of an arithmetic series is 504. The first term is 9. Find the 12^th term. Answer.12th term = 42.	16. The sum of the first 6 terms of an arithmetic series is 66 while the sum of the next six terms of the series is 282. (i) Find the first term of the series and the common difference. (ii) Find the sum of terms T₁₃ to T₁₈. Answer. (i) 1st term = -4. Difference = 6. (ii) Sum of 18 terms is 846 so Sum (13-18) = 498.
Using the sum to find the number of terms.	17. The first and last terms in an arithmetic progression are 10 and 60. The sum of the series is 3535. How many terms in the series? Answer.101 terms.	18. The sum of a number of terms of an arithmetic series is 100. The first term is -25 and the last term is 35. Find the number of terms in the series. Answer.n = 20 terms.
	19. The sum of an arithmetic series can be expressed as S_n = 2n². Find the 10th term in the series. Answer.T₁₀ = 38.	20. The first three terms of an arithmetic series are 6, 10 and 14. Calculate the number of terms needed to give a sum of 390. Answer.n = 13 terms.