Arithmetic/Geometric sequences

[jai6638]

hey... I'd appreciate it if you could verify my answers..

Q1) Write a rule for the nth term of the arithmetic sequence 1,6,11,16... Then find a10.

A1) An=A1 + (n-1)d
An=1+ (n-1)5

A10=1+(10-1)5
A10=1+45
A10=46
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Q2) Find the sum of the first 20 terms of the arithmetic series:

.5+1.1+1.7+2.3+2.9+...

A2) An=a1+(n-1)d
A20= .5 + (20-1).6
a20= 11.4+.5 = 11.9

Sn= n (a1+an/2)
sn= 10 (.5+11.9/12)
Sn=124
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Q3) Write a rule for the nth term of the geometric sequence 3,15,75,375,... then find a12

a3) An=A1r^(n-1)
an=(3)(5)^(n-1)

A12=(3)(5)^(12-1)
a12=(3)(5)^11
a12=146,484,375
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Q4) Your father wants to make a deal with you. He will giev you five cents to clean the garage. He will then double the amount daily for each day this month that you keep the garage tidy. Today is August 15. Is it a good idea to accept these conditions? How much will you earn on August 31?

A4) C= .5
n=number of days

Expression: an= 2c+n
an=2(.5)+n
an= n+1

A15=15+1
a15=16

He would earn $16 on august 31st.
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I do not know the answers for the following questions.

Q5) Write the next term in the sequence. Then write a rule for the nth term.

a) 7,-8,9,-10

Aa)
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b) (1/2), (-1/4), (1/8), (-1/16)..

Ab) i was thinking the rule could probably be 1/2^(n+2) but the problem is that i won't get the negative sign..
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Any help is much appreciated.

 

[Data]

1 is correct, the answer for 2 is correct but I don't know where you got your steps from (probably just a typo), 3 is correct, 4 is wrong (at least the way I read it - he's doubling the amount he paid you the previous day on each successive day).

For 5, how can you get an extra (-1) alernating between terms? Any ideas? HINT: If the general term of a sequence is given by $(-1)^n$ then what are the terms?

 

[jai6638]

For 5, how can you get an extra (-1) alernating between terms? Any ideas? HINT: If the general term of a sequence is given by then what are the terms?

although I do somewhat understand what your saying, i can't seem to think of a way to apply it to 5a and 5b... For example, in 5a, 9,7,-8,-10... i don't see any common difference... they are not common multiples or anything so how would i make the rule?

EDIT: for 5b) i think i found the answer.. its (-1)/(-2)^(n+2).. therefore the next term would be (1/32) .. is that correct?

4 is wrong (at least the way I read it - he's doubling the amount he paid you the previous day on each successive day).

A4) will it be, c+2c*n= c+2cn= n ( c+2c)?

 

[VietDao29]

For 5b/ Your answer is correct. But it would be better if you write it like:
$$A_{n} = A_{1} r^{n - 1} = \frac{1}{2} \left(-\frac{1}{2}\right)^{n - 1}$$
For Q4. It's also a geometric sequence:
You have $A_{1} = 5 \mbox{ cents}$. You have r = 2. So can you find the 17th term? (The day 31th August is the 17th term of the se quence) (I wonder if any dads will be as generous as the dad in this problem ).
Viet Dao,

 

[Data]

For 5a, first think about the sequence $$7, 8, 9, \cdots$$. How would you express the general term for that? Once you've thought about it, can you see a way to use the same trick I pointed out for 5b to get the sequence the question asks for?