Sequences of geometry?

Question

In exercises 1-4, three of the five variables a1, an, r, n, and Sn are given. Find the two variables not given.

1.

a1 = 10, r = 3, n = 5

2.

a1 = 4, an = 324, r = 3

3.

an = 24, r =2/3 , n = 4

4.

Sn = 1022, r = 2, n = 9

5.

A city has a population of 20,000 people in the year 2000. If the population increases 5% per year, what will the population be in 2010? ( You are not summing the population over 2000-2010. Find the population in the year 2010, use a1rn to solve.)

6.

At the age of 21, Cyndi began receiving yearly payments from a trust fund. On each succeeding birthday, she received twice as much as in the preceding year. If she had received a total of $303,800 by age 25, how much did she receive at age 21?

Answer 1

Hi,

1. a1 = 10, r = 3, n = 5

an = a1*r^n

an = 10*3^5

an = 2,430 <==ANSWER

a(1 - r^n)

------------ = Sn

1 - r

10(1 - 3^5)

------------ = Sn

1 - 3

Sn = 1,210 <==ANSWER

2. a1 = 4, an = 324, r = 3

an = a1*r^n

324 = 4*3^n

81 = 3^n

3^4 = 3^n

n = 4 <==ANSWER

a(1 - r^n)

------------ = Sn

1 - r

4(1 - 3^4)

------------ = Sn

1 - 3

Sn = 160 <==ANSWER

3. an = 24, r =2/3 , n = 4

an = a1*r^n

24 = a1*(2/3)^4

24 = a1*(16/81)

a1 = 243/2 <==ANSWER

a(1 - r^n)

------------ = Sn

1 - r

243/2(1 - (2/3)^4)

------------------------ = Sn

1 - 2/3

Sn = 585/2 <==ANSWER

4. Sn = 1022, r = 2, n = 9

a(1 - r^n)

------------ = Sn

1 - r

a(1 - 2^9)

------------ = 1022

1 - 2

a(1 - 512)

------------ = 1022

1 - 2

-511a

--------- = 1022

-1

511a = 1022

a1 = 2 <==ANSWER



5. A city has a population of 20,000 people in the year 2000. If the population increases 5% per year, what will the population be in 2010? ( You are not summing the population over 2000-2010. Find the population in the year 2010, use a1rn to solve.)

an = a1*r^n

an = 20,000*1.05^10

an = 32,578 <==ANSWER

6. At the age of 21, Cyndi began receiving yearly payments from a trust fund. On each succeeding birthday, she received twice as much as in the preceding year. If she had received a total of $303,800 by age 25, how much did she receive at age 21?

x + 2x + 4x + 8x + 16x = 303,800

31x = 303,800

x = $9,800 <==ANSWER

I hope that helps!! :-)