need help with arithmetic geometric squences,?

Question

I have several problems I am trying to work out I need to find using the nth term of arithmetic seq. what the 101st term and also find the sum of the 20th term and then the 30th term> understand that the ratio is +2 in this sequence 1,2.4,8 but I don understand how to figure what the term is. really. and the nth term i dont get at all.
I also have to figure the same thing different numbers in the geometric sequence 1,1/2,1/4,1/8 .
but I don't know using the nth term of geometric seq. what is the 24th term. and using the same seq. finding the sum of the first 10 terms and them the 12 term rounded to the 4 decimal.
and how do you tell what whole number will the sum be smaller than. I would really appreiciate if someone could talk me through how to determine these things , like explaining where each number comes from. thanks

Answer 1

for 1,2,4,8
since the numbers are 1,2,4,6,8, this is a geometric sequence, not a arithmetic sequence

Geometric Sequence Formula
an = a1 * r^(n - 1)

a1 = 1
r = 2

The pattern is 2^(n - 1)

a(101) = 2^(101 - 1)
a(101) = 2^100
Since the answer is so long, i will leave the answer at 2^100

As for the summation(sum of terms)

Summation formula for a Geometric Sequence

Sn = (a1(1 - r^n))/(1 - r)
S(20) = (1(1 - 2^(20)))/(1 - 2)
S(20) = (1 - 2^20)/-1
S(20) = 1048575

S(30) = -(1 - 2^30)
S(30) = 1073741823

ANS:
a(101) = 2^100
S(20) = 1048575
S(30) = 1073741823

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Patter is like the one before

an = 2^(-(r - 1))
an = 2^(-r + 1)

a(24) = 2^(-24 + 1)
a(24) = 2^(-23)
a(24) = 1/8388608
Can't round this to 4 decimal spaces, because its they are all 0s

S(10) = -1/(1 - 2^n)
S(10) = -1/(1 - 2^10)
S(10) = -1/(-1023)
S(10) = 1/1023
S(10) = .0010

S(12) = -1/(1 - 2^12)
S(12) = -1/(-4095)
S(12) = 1/4095
S(12) = .0002

ANS :
a(24) = 1/8388608
S(10) = .0010
S(12) = .0002

Answer 2

If the series is 1,2,4,8, the common ratio is 2.
1 is the first term, 2 is the second term, 4 is the third, etc.
The formula to find the nth is term is n = a*r^n-1
where n is the nth term
a is the first term (1 in this particular problem)
r is the common ratio (2 in this problem)

For example if you wanted to find the 10th term in the series:
n = 1*2^10-1
n = 1*2^9
n= 1*512 = 512.
So, the tenth term would be 512

Answer 3

arithmetic sequence is one in which the difference between any term and its predecessor is constant and is known as the common difference
so a typical arithmetic sequence will be
a,a+d,a+2d,a+3d,a+4d etc note a+d-a=d;a+2d-(a+d)=d;a+3d-(a+2d)=d and so on
a1 being the first term can be written as a+0d
a2=a1+1d
a3=a2+d=a1+d+d=a1+2d
a4=a3+d=a1+2d+d=a1+3d
so an=a1+(n-1)d
an is called the n the term
whatever term you want that will be n
for example if you want the 10th term n=10
a10=a1+(10-1)d=a1+9d

sum of the n terms Sn=a+a+d+a+2d+a+3d+.......+a+(n-1)d
=n/2[2a+(n-1)d]

the geometric sequence will have a common ratio 'r'.
that is the quotient of ant term divided by its predecessor will be constant
the general form will be
a,ar,ar^2,ar^3.......ar^(n-1)
Sn=a[r^n-1/r-1]
if you post some specific sums and see us solve them you will get a better idea
good luck

Answer 4

i don't know.