How do i do this?? sequencing?

Question

1. The third term of an arithmetic sequence is 9 and the seventh term is 31. Find the sum of the first twenty-two terms.


2. How many terms of the arithmetic sequence 2,4,6,8, ... add up to 60,762?
PLease help IM frustrated!!!!

Answer 1

Remember the sum of n terms of an arithmetic sequence with first term a and difference d is:
(n/2)(2a + (n-1)d).

1)
The difference between the seventh and third terms is (31-9) = 22. Since they are four terms apart, the common difference is d = 22/4 = 5.5.
The first term will be two differences behind the third, ie 9 - 2*d = -2.

Substitute into the sum formula:
22/2 * (2*-2 + 21*5.5) = 1226.5.

2)
Substitute into the sum formula:
60762 = (n/2)(2*2 + (n-1)*2)
= (n/2)(4 + 2n - 2)
= 2n + n^2 - n
= n^2 + n.
So you have n^2 + n - 60762 = 0, so (n-246)(n+247) = 0, so n = 246 terms.

Answer 2

an = a1 + (n - 1)d

a3 = a1 + (3 - 1)d
9 = a1 + 2d
a1 = 9 - 2d

a7 = a1 + (7 - 1)d
31 = a1 + 6d
a1 = 31 - 6d

9 - 2d = 31 - 6d
4d = 22
d = (11/2)

a1 = 31 - 6(11/2)
a1 = 31 - (66/2)
a1 = 31 - 33
a1 = -2

now that we know the first term, you can use this formula

Sn = (n/2)(a1 + an)

a(22) = -2 + (22 - 1)(11/2)
a(22) = -2 + 21(11/2)
a(22) = -2 + (231/2)
a(22) = (-4/2) + (231/2)
a(22) = (227/2) or 113.5

S(22) = (22/2)(-2 + 113.5)
S(22) = 11(111.5)
S(22) = 1226.5

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an = a1 + (n - 1)d
Sn = (n/2)(a1 + an)

We can already tell that the pattern is 2

an = 2 + (n - 1)2
an = 2 + 2(n - 1)
an = 2(1 + (n - 1))
an = 2(1 + n - 1)
an = 2(n)
an = 2n

60762 = (n/2)(2 + 2n)
60762 = (n/2)(2(1 + n))
60762 = (2n/2)(1 + n)
60762 = n(n + 1)
n^2 + n = 60762
n^2 + n - 60762 = 0

Using the quadratic formula

n = (-1 ± sqrt(1 + 243048))/2
n = (-1 ± sqrt(243049))/2
n = (-1 ± 493)/2
n = (-494/2) or (492/2)
n = -247 or 246

ANS : The 246th term

Answer 3

I would recommend deriving what you need:

1. The difference between element is some integer multiple of d, the consequitive difference:

(7-3)d = 31-9
4d = 22
d=22/4

So, the 22nd term would be
(22-7)d = T22-31
T22 = 15d+31

Need to find the first term:
T1 = T3-2d = 9-2d
then the sum is

S=

Answer 4

that sounds hard