Mathematics quiz!?

Question

Find the n'th term of the following sequences:

a) 4,11,22,37,...

b) 6,17,34,57, ...

c)6, 17, 40, 81, 146, ...

d)2, 9/8, 28/27, 65/64, 126/125, ...

e)6, 15/2, 92/9, 105/8, 402/25, ...

The first person getting the correct answers for every question gets 10 points.
Another point, if someone answered without giving answers, I hope you people would report that person.
Have fun~!

n'th, not 9th. Example: 1,3,5,7,9...
The n'th number is 2n+1.

Answer 1

Solving the n-th term is all about interpolating points into a quadratic.

S(n) = a(n)^2 + b(n) + c

All we need to do is solve this system of equations:

n1 = a + b + c
n2 = 4a + 2b + c
n3 = 9a + 3b + c

where
n1 = the first number
n2 = the second number
n3 = the third number

I won't show you the gory details, but the solution for a, b, and c are:

a = (n1 - 2n2 + n3)/2
b = (-5n1 + 8n2 - 3n3)/2
c = 3n1 - 3n2 + n3

And all we have to do is plug in those values to solve the sequence. Let's give it a shot.

a) 4, 11, 22, 37

a = (4 - 2(11) + 22)/2 = (4 - 22 + 22)/2 = 4/2 = 2
b = (-5(4) + 8(11) - 3(22))/2 = (-20 + 88 - 66)/2 = 1
c = 3(4) - 3(11) + 22 = 12 - 33 + 22 = 1

Therefore, our formula is S(n) = 2n^2 + n + 1
S(4) = 2(4)^2 + 4 + 1 = 32 + 4 + 1 = 37

b) 6, 17, 34, 57

a = (6 - 2(17) + 34)/2 = 6/2 = 3
b = (-5(6) + 8(17) - 3(34))/2 = (-30 + 136 - 102)/2 = 2
c = 3(6) - 3(17) + 34 = 18 - 51 + 34 = 1

Therefore, S(n) = 3(n)^2 + 2(n) + 1
S(4) = 3(4)^2 + 2(4) + 1 = 3(16) + 8 + 1 = 57

c) 6, 17, 40, 81, 146, ...

a = (6 - 2(17) + 40)/2 = 22/2 = 11
b = (-5(6) + 8(17) - 3(40))/2 = -7
c = 3(6) - 3(17) + 40 = 7

S(n) = 11x^2 - 7x + 7
S(4) = 11(4)^2 - 7(4) + 7 = 176 - 28 + 7 = 155

Okay, (c) through (e) don't follow the same quadratic pattern as (a) and (b). And they'll take forever to solve for, so I won't, but I do know the method.

Answer 2

a) 9th term is 172.

The successive terms have differences that increase by 4.
So the successive terms of the sequence of successive
difference is the constant 4. To show this, write each
sequence in a line, the successive differences of a sequence
under this sequence, and so. You should get:
4 11 22 37
7 11 15
4 4
By continuing the terms in each line one by one,starting from
the last to up to the first you get the next terms of the first
required sequence.
The way to do this is:
1) The last sequence is the constant 4, so it will always be 4.
2) The next term in a line is the sum of the current term and the
term below it.
By continuing this way, you should get 172 as the 9th term.


b) 9th term is 262.

Use the same idea as that for the previous
part. Note that the constant difference is 6.


c) 9th term is 766

Use the previous way. Now, there will be 4 sequences
including the constant sequence which is 6.


d) 9th term is 730/729

This sequence has the general formula (1+n^3)/n^3. So just
plug 9 to find the term.


e) 9th term is 2300/81

If you make the denominator n^2 by multiplying the numerator
and denominator by the required number, you find that the
numerators, when treated as a sequence in itself, can be
treated as the first 3 parts. If you apply to the
sequence the same idea applied to the first 3, you get the
numerator of the required term. The denominator is 9^2.

Answer 3

Sorry, I didn't see the qns properly last time, anyway,
Answers:
a)2n^2+n+1
b)3n^2+2n +1
c)n^3+4n+1
d)(n^3+1)/n^3
e)((3n^2-6n+5)^2+(3n^2-6n+5))/n squared
(divided by n squared)
Gimme my points

Answer 4

a) 2*n^2 + n +1

b) 3*n^2 + 2*n +1

c) n^3 + 4*n +1

d) (n^3 +1 )/(n^3)

e) 3*n + 1+(2/(n^2))

Answer 5

a: 56,79,106,147,182= 182=the 9th term
b:idk
idk the rest i tried....