Arithmetic series Problem set B Worked solutions

Find each sum.

1. (a) 3 + 10 + 17 + … (20 terms)

   • \(S_n = \frac{n}{2}(2u_1 + (n-1)d)\)
   • \(S_{20} = 10(2(3) + 19(7)) = 10(139)\)
   • \(S_{20} = 1390\)

2. (b) 50 + 45 + 40 + … (30 terms)

   • \(S_n = \frac{n}{2}(2u_1 + (n-1)d)\)
   • \(S_{30} = 15(2(50) + 29(-5)) = 15(-45)\)
   • \(S_{30} = -675\)

3. (c) The first 60 even numbers

   • The even numbers are 2, 4, 6, …: \(u_1 = 2,\ d = 2\)
   • \(S_{60} = 30(2(2) + 59(2)) = 30(122)\)
   • \(S_{60} = 3660\)

An arithmetic series has first term 3, and the sum of its first 12 terms is 300. Find the common difference.

• \(S_{12} = 6(2(3) + 11d) = 300\)
• \(6 + 11d = 50\)
• \(d = 4\)

Find each sum.

1. (a) 15 terms, first term 4, last term 88

   • \(S_n = \frac{n}{2}(u_1 + u_n)\)
   • \(S_{15} = \frac{15}{2}(4 + 88)\)
   • \(S_{15} = 690\)

2. (b) 2 + 5 + 8 + … + 101

   • \(d = 3:\ 2 + 3(n-1) = 101\)
   • \(n = 34\)
   • \(S_{34} = 17(2 + 101) = 1751\)

Answer each part.

1. (a) How many terms of 4 + 10 + 16 + … are needed for the sum to first exceed 690?

   • \(S_n = \frac{n}{2}(8 + 6(n-1)) = n(3n + 1)\)
   • \(S_{15} = 690\) (not more than 690), \(S_{16} = 784\)
   • \(n = 16\)

2. (b) The sum of the first \(n\) terms of 5 + 9 + 13 + … is 779. Find \(n\).

   • \(S_n = \frac{n}{2}(10 + 4(n-1)) = n(2n + 3)\)
   • \(n(2n + 3) = 779\)
   • \(2n^2 + 3n - 779 = 0,\ n = 19\) (the other root is negative)

The 5th term of an arithmetic series is 23 and the 11th term is 47. Find the sum of the first 14 terms.

• \(u_1 + 4d = 23\) and \(u_1 + 10d = 47\)
• \(d = 4,\ u_1 = 7\)
• \(S_{14} = \frac{14}{2}\,[2(7) + 13(4)] = 462\)

Pipes are stacked with 30 in the bottom row and one fewer in each row above. The top row has 11 pipes. How many pipes are there?

• \(30 - (n-1) = 11\)
• \(n = 20\) rows
• \(S_n = \frac{n}{2}(u_1 + u_n)\)
• \(S_{20} = 10(30 + 11) = 410\) pipes

Tomas saves €30 in the first month and €4 more each month than the month before. (a) How much has he saved in total after 20 months? (b) In which month does his total first pass €2000?

• \(S_{20} = 10(2(30) + 19(4)) = 1360\)
• (a) €1360
• \(S_n = \frac{n}{2}(60 + 4(n-1)) = 2n^2 + 28n\)
• \(S_{25} = 1950,\ S_{26} = 2080\)
• (b) Month 26

The sum of the first \(n\) terms of a sequence is \(S_n = 4n^2 - n\). Find the first term and a rule for \(u_n\).

• \(u_1 = S_1 = 4 - 1 = 3\)
• \(u_n = S_n - S_{n-1} = 4n^2 - n - 4(n-1)^2 + (n-1)\)
• \(u_n = 4(2n - 1) - 1 = 8n - 5\)

Find the sum of the multiples of 9 from 9 to 360.

• \(u_n = 9n\)
• \(9n = 360,\ n = 40\)
• \(S_n = \frac{n}{2}(u_1 + u_n)\)
• \(S_{40} = 20(9 + 360) = 7380\)