Sequences and Series

Here you will find a concise collection of arithmetic and geometric sequences and series practice problems. Visit this page directly by visiting hunkim.com/sequences or hunkim.com/seq

Overview:

• Arithmetic Sequences and Series
• Geometric Sequences and Series

Arithmetic Sequences and Series

• Arithmetic sequence: t_n=t_1+(n-1)d
• Arithmetic series:
  • S_n=\frac{n}{2}(t_1+t_n)
  • S_n=\frac{n}{2}\left(2t_1+(n-1)d\right)

1. 2, 5, 8, 11, ...
   1. Express this pattern as a line equation
      Solution
      y=3x-1
      
      
   2. Express this pattern using the arithmetic sequence formula t_n = t_1 + (n-1)d
      Solution
      t_n=2+(n-1)(3)
      
      
   3. Justify why this formula makes sense
      Solution
      We start at the value of the first term. We then add multiples of the common difference d to achieve the unknown term t_n
      
      
   4. Find the 100th term of this sequence
      Solution
      299
      
      
2. 20, 17, 14, ... -37. How many terms are in this sequence?
   Solution
   20
   
   
3. \frac{2}{3} and 1 \frac{1}{2} are the first two terms of an arithmetic sequence. Find the 4th term
   Solution
   \frac{19}{6}
   
   
4. 3.\bar{3} and 4.\bar{6} are the 4th and 5th terms of an arithmetic sequence. Find the first term
   Solution
   -\frac{2}{3}
   
   
5. The third term of an arithmetic sequence is 11 and the 15th term has a value of 47. Find the value of the 30th term.
   Solution
   62
   
   
6. Enrichment: Understanding the arithmetic series formula
   1. Explain why S_n=\frac{n}{2}\left(2t_1+(n-1)d\right)
      Solution
      See proof here
      
      
   2. Why is this formula equivalent to S_n=\frac{n}{2}(t_1+t_n)
      Solution
      Substitute t_n=t_1+(n-1)d into the other formula
      
      
7. 7, 4, 1, -2, ..., -50. Find the sum of these terms
   Solution
   -430
   
   
8. Express the sum of 1+2+3+4+5+...+n as a formula in terms of n
   Solution
   S_n=\frac{n}{2}\left(1+n\right)
   
   
9. Find the sum of all the multiples of 3 between 50 and 5000
   Solution
   4165425
   
   
10. An arithmetic sequence has first term 3 and common difference 6. If the sum of all terms is 4800, how many terms are in the sequence?
    Solution
    40
    
    
11. -10, -6, -2, 2, ... Find the least number of terms so that the sum of the series is greater than 960
    Solution
    26
    
    
12. The sum of the first n terms of an arithmetic sequence is
    S_n=5n^2-3n. Find the nth term t_n
    Solution
    10n-8
    
    
13. What is the sum of all three-digit numbers which are multiples of 6 but not 9?
    Solution
    54900
    
    
14. S_{10}=285 and S_{14}=539. Find S_5, the sum of the first 5 terms in an arithmetic series
    Solution
    80
    
    
15. Enrichment:
    1. Given t_n=3n-2 simplify an expression for t_{n+1}-t_n
       Solution
       3
       
       
    2. t_1=1 and t_2=1. Given t_n=t_{n-1}+t_{n-2}, find t_4
       Solution
       3
       
       
16. Sigma notation:
    1. \sum_2^5{x}
       Solution
       14
       
       
    2. \sum_0^2 \left(x^2-3x\right)
       Solution
       -6
       
       
    3. \sum_{-1}^2{\frac{x}{3}}
       Solution
       \frac{2}{3}
       
       
    4. \sum_1^4{\frac{2\pi r}{e}}
       Solution
       \frac{20\pi}{e}
       
       
    5. Evaluate \sum_2^4\left(3^r-1\right)
       Solution
       116
       
       
    6. T_n=\sum_0^n\left(r^2-2r\right). Find T_2
       Solution
       -1
       
       
17. How many terms does \sum_{5}^{100}r have?
    Solution
    96
    
    
18. Find the sum of the numbers between 500 and 2000 that are multiples of 3
    Solution
    624750
    
    
19. Write in sigma notation:
    1. 10+7+4+1
       Solution
       \sum_1^4(-3n+13)
       
       
    2. \frac{2}{3}+\frac{3}{4}+\frac{4}{5}+\frac{5}{6}
       Solution
       \sum_2^5\frac{r}{r+1} or \sum_3^6\frac{r-1}{r}
       
       
    3. Challenge: \frac{2x}{\pi}+\frac{5x}{2\pi}+\frac{8x}{3\pi}
       Solution
       \sum_1^3\frac{(3n-1)x}{n\pi}
       
       
20. S_1=1, S_2=1+3, S_3=1+3+5, S_4=1+3+5+7... Evaluate S_n in terms of n
    Solution
    S_n=n^2
    
    
21. Challenge:
    1. Challenge: 3\sqrt{3} and 2\sqrt{27} are respectively the second and the thrid terms of an arithmetic sequences. Find the value of the 5th term
       Solution
       12\sqrt{3}
    2. The first four terms of an arithmetic sequence are:
       4, 3b-a, 4a+b-1, 5b-a. Find a and b
       Solution
       a=2, b=3
    3. 2a, a+2b, 5b-2a are the first three terms of an arithmetic sequence
       1. Find an expression for the 5th term
          Solution
          -2a+8b
       2. Express a in terms of b
          Solution
          a=\frac{b}{2}
       3. Given t_4=11, find a
          Solution
          1
          
          
    4. A circular is cut into 8 sectors whose angles are in an arithmetic sequence. The angle of the largest sector is triple the angle of the smallest sector. Find the angle of the largest sector.
       Solution
       67.5^\circ
       
       
    5. The ratio of the 4th term to the 10th term of an arithmetic sequence is 11:23. If each term of this sequence is positive and the product of the first term and the third term is 45, find the sum of the first 100 terms of the sequence.
       Solution
       104000
       
       
    6. An arithmetic sequence has the first term \ln a and a common difference \ln 2. The 14th term in the sequence is 4\ln 16. Find the value of a
       Solution
       8

Geometric Sequences and Series

• Geometric sequence: t_n=t_1r^{n-1}
• Geometric series: S_n=\frac{t_1(1-r^n)}{1-r}
• Infinite geometric series: S_\infty=\frac{a}{1-r}

1. 3, 6, 12, 24, ... Find the 10th term using the geometric series formula: t_n=t_1 r^{n-1}
   Solution
   1536
   
   
2. 1024, 512, 256, ... \frac{1}{4}. How many terms in this sequence?
   Solution
   13
   
   
3. In a geometric sequence, the 2nd term is 28 and the 5th term is 1792. Find t_1
   Solution
   7
   
   
4. 0.2+0.002+0.00002+...
   1. Find t_1
      Solution
      0.2
      
      
   2. Find r
      Solution
      0.01
      
      
5. 3, 12, 48, 5y+7, ... Find y
   Solution
   37
   
   
6. The first three terms of a geometric sequence are:
   t_1=0.32, t_2=1.28, and t_3=5.12.
   1. Find the value of r
      Solution
      4
      
      
   2. Find the value of S_5
      Solution
      109.12
      
      
   3. Find the least value of n such that S_n>30,000
      Solution
      10
      
      
7. You jog to school. During the first minute you jog 60 metres. In each subsequent minute you walk 95% of the distance jogged during the previous minute. The distance between your house and school is 1000 metres. You leave your house at 8 am. What time will it be when you arrive at school?
   Solution
   8:35 am
   
   
8. A ball is dropped vertically from a height of 12 m. After each bound it rises to 80% of its previous height
   1. What height does the ball reach after the 4th bounce?
      Solution
      About 4.92 m
      
      
   2. After how many bounces will the ball reach an approximate height of 10 cm?
      Solution
      21 or 22 bounces
      
      
   3. What is the total vertical distance travelled by the ball as soon as the ball bounces 10 times?
      Solution
      About 95.1 m
      
      
   4. What is the total distance travelled by the ball if it bounces indefinitely?
      Solution
      About 108 m
      
      
9. You inherited $5000 at the beginning of the year 2020.  Assuming a growth rate of 10%, how much money will you investment grow to be by the end of the year 2070?
   Solution
   About $586,954.26
   
   
10. The sum of an infinite geometric sequence is 25. The common ratio is \frac{4}{5}. Find the first term
    Solution
    \frac{125}{4}
    
    
11. Three consecutive terms of a geometric sequence are:
    x-5, x-2, x+4. Find the value of the first of these three terms
    Solution
    3
    
    
12. \sum_{2}^{10}3^r
    1. Find r
       Solution
       3
    2. Evaluate
       Solution
       265716
       
       
13. S_1=1+k, S_2=4k+2, S_3=8k+5
    1. Express t_1 in terms of k
       Solution
       t_1=1+k
       
       
    2. Find t_3
       Solution
       4k+3
       
       
14. t_n=n3^n
    1. Find t_1
       Solution
       3
       
       
    2. Simplify \frac{t_{n+1}}{t_n}
       Solution
       \frac{3n+3}{n}
       
       
15. 9, 3, 1, .... Find S_\infty
    Solution
    \frac{27}{2}
    
    
16. 4, 6, 9, \frac{27}{2}, .... Find S_\infty
    Solution
    No solution, this series diverges
    
    
17. See figures below:
    Fig 1:
    
    
    Fig 2:
    
    
    Fig 3:
    
    1. If the side length of figure 1 is 16 metres what is the shaded region of figure 10?
       Solution
       \approx2.44\times10^{-4}
       
       
    2. If the side length of figure 1 is k metres, what is the area of the non-shaded region as n\to\infty?
       Solution
       \frac{2k^2}{3}
       
       
18. Challenge:
    1. Solve \sum_{k=1}^{\infty} \left(3^{3-k} \log x\right)=\frac{200}{9}
       Solution
       10^{\frac{400}{81}}
       
       
    2. A book starts at page 1 and is numbered on every page.
       1. How many page digits in total are there on the first dozen pages?
          Solution
          15
          
          
       2. If the total number of digits used is 252, how many pages are in the book?
          Solution
          120
          
          
    3. Evaluate \sum_{n=0}^{n=\infty} \frac{2^n+4^n}{6^n}
       Solution
       \frac{9}{2}
       
       
    4. The first three terms of a geometric sequence are \log x^9, \log x^3, \log x, for x>0. Find the common ratio
       Solution
       \frac{1}{3}