Pre Cal 11 Help

Question

1.) Is this arithmetic series convergent or divergent? 1/4, 3/4, 5/4

2.)An infinite geometric series converges to 12. Determine t3 if t1 = 3.

3.) The sum of nine consecutive positive integers is 99. Determine the largest of these integers.

4.)What value(s) of x will make the following geometric series converge?

1. 4 + 12x + 36x^2 + 108x^3 + ⋯

5.)While training for a race, a runner increases her distance by 10% each day. If she runs 2 Km on the first day, what will be the total distance that she has run after 30 days of training?

Please answer all or most of them, showing work. Thank you so much!

Answer 1

1.) Is this arithmetic series convergent or divergent? 1/4, 3/4, 5/4

by simple inspection , you can tell R of the arithmetic series is 2/4. adding 1/2 to infinity, will go on to infinity.

diverge to infinity is the answer.

2.)An infinite geometric series converges to 12. Determine t3 if t1 = 3.

S=t₁+t₂+t₃+.....+t_∞

₁₂₌t₁₊t₁(r)¹+t₁(r)²t₁(r)³+.....+t_∞

₁₂₌t₁(1+r¹+r²+...+r^∞)

12=3(1+r¹+r²+...+r^∞)

4=(1+r¹+r²+...+r^∞)

4=1/(1-r)

4-4r=1

r=4/3

t₃=t₁(r)²=3*(4/3)²=16/3=5.333333

3.) The sum of nine consecutive positive integers is 99. Determine the largest of these integers.

a1+a2+a3=99

3a2=99

a1=31

a2=33

a3=34

4.)What value(s) of x will make the following geometric series converge?

1. 4 + 12x + 36x^2 + 108x^3 + ⋯

For any geometric series to converge, R needs to be between 0 and 1.

In this case 0<x<1

5.)While training for a race, a runner increases her distance by 10% each day. If she runs 2 Km on the first day, what will be the total distance that she has run after 30 days of training?

S=t₁+t₂+t₃+.....+t₁₂

_S=t₁₊t₁(r)¹+t₁(r)²t₁(r)³+.....+t₁(r)³⁰

S=2(1-1.1³⁰)/(1-1.1)

S=2(164.494)

S=329km