Sequences and Series — Practice

Question 1

Write the first three terms of the sequence whose nth term is an = 3n - 1.

Accepted Answer

a1=3(1)-1=2. a2=3(2)-1=5. a3=3(3)-1=8. Answer: The first three terms are 2, 5, 8.

Question 2

Find the 6th term of the G.P. 2, 6, 18, ...

Accepted Answer

a=2, r=6/2=3. a6=ar^5=2(3)^5=2(243)=486. Answer: The 6th term is 486.

Question 3

In a G.P., the 3rd term is 18 and the 6th term is 486. Find the common ratio and the first term.

Accepted Answer

a3=ar^2=18, a6=ar^5=486. Dividing: r^3=486/18=27, so r=3. Then a(9)=18, so a=2. Answer: The common ratio is 3 and the first term is 2.

Question 4

Find the sum of the first 8 terms of the G.P. 1, 2, 4, 8, ...

Accepted Answer

a=1, r=2, n=8. S8 = 1(2^8-1)/(2-1) = 255/1 = 255. Answer: The sum of the first 8 terms is 255.

Question 5

Find the sum of the first 6 terms of the G.P. 3, -6, 12, -24, ...

Accepted Answer

a=3, r=-2, n=6. S6 = 3((-2)^6-1)/(-2-1) = 3(64-1)/(-3) = 3(63)/(-3) = -63. Answer: The sum of the first 6 terms is -63.

Question 6

Find the sum to infinity of the G.P. 4, 2, 1, 1/2, ...

Accepted Answer

a=4, r=1/2. Since |r|<1: S(infinity) = 4/(1-1/2) = 4/(1/2) = 8. Answer: The sum to infinity is 8.

Question 7

Express the recurring decimal 0.272727... as a fraction in lowest terms.

Accepted Answer

Write it as a G.P.: 27/100 + 27/10000 + ... with a=27/100, r=1/100. S(infinity) = (27/100)/(1-1/100) = (27/100)/(99/100) = 27/99 = 3/11. Answer: 0.272727... equals 3/11.

Question 8

Find the geometric mean of 4 and 25.

Accepted Answer

G.M. = sqrt(4 x 25) = sqrt(100) = 10. Answer: The geometric mean of 4 and 25 is 10.

Question 9

Insert three geometric means between 2 and 32.

Accepted Answer

a=2, and 32 is the 5th term (a, G1, G2, G3, 32): 32=2.r^4, so r^4=16, r=2. The means are 2(2)=4, 2(2)^2=8, 2(2)^3=16. Answer: The three geometric means are 4, 8, and 16.

Question 10

Find the A.M. and G.M. of 9 and 16, and verify that A.M. >= G.M.

Accepted Answer

A.M. = (9+16)/2 = 12.5. G.M. = sqrt(9x16) = sqrt(144) = 12. Since 12.5 >= 12, the relationship A.M. >= G.M. holds, as it must for any two positive numbers. Answer: A.M.=12.5, G.M.=12, and A.M. >= G.M. is confirmed.

Practice — Sequences and Series

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