Arithmetic sequence questions?!?! 10 pts for best answers?

Question

Find the first term of the arithmetic sequence having the data given.
1.) avn = 48, Svn = 546, n = 26
2.) Svn = 781, d = 3, n = 22
3.) The third term of an arithmetic sequence is 9 and the seventh term is 31. Find the sum of the first twenty-two terms.
4.) How many terms of the arithmetic sequence 2,4,6,8, ... add up to 60,762?

Answer 1

Question 1:

Given: a(n) = 48, s(n) = 546, n = 26



Use this formula.

S(n) = (n/2)(a(1) + a(n))

546 = (26/2)(a(1) + 48)

546 = 13a(1) + 624

546 - 624 = 13a(1)

-6 = a(1) (Answer)



Question 2:

Given: S(n) = 781, d = 3, n = 2



Remember that: a(n) = a(1) + (n - 1)d



S(n) = (n/2)(a(1) + a(n))

S(n) = (n/2)(a(1) + a(1) + (n - 1)d)

781 = (22/2)(2a(1) + (22 - 1)(3))

781 = 22a(1) + 693

781 - 693 = 22a(1)

88 = 22a(1)

4 = a(1) (Answer)



Question 3:

Given: a(3) = 9, a(7) = 31



If a(3) and a(7) were consecutive terms, the common difference will be 4 times bigger, since you must add the common difference 4 times to a(3) to reach a(7). Let a(3) be a(1) and a(7) be a(2).



a(n) = a(1) + (n - 1)d

31 = 9 + (2 - 1)d

31 - 9 = d

22 = d



So the common difference in this problem is 22/4 or 11/2. Now, let's find the first term of the sequence. Let n = 3.



a(n) = a(1) + (n - 1)d

9 = a(1) + (3 - 1)(11/2)

9 = a(1) + 11

9 - 11 = a(1)

-2 = a(1)



Find the 22nd term.

a(n) = a(1) + (n - 1)d

a(22) = -2 + (22 - 1)(11/2)

a(22) = -2 + 231/2

a(22) = 227/2



Finally, find the sum of the first 22 terms.

S(n) = (n/2)(a(1) + a(n))

S(n) = (22/2)(-2 + 227/2)

S(n) = (11)(223/2)

S(n) = 2453/2 (Answer)



Question 4:

Given: a(1) = 2, d = 2, S(n) = 60762



Using the formula for the sum of an arithmetic sequence, form the first equation.

S(n) = (n/2)(a1 + a(n))

60762 = (n/2)(a1 + a1 + (n - 1)d)

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

60762 = (n/2)(4 + 2n - 2)

60762 = (n/2)(2 + 2n)

60762 = n(1 + n)

60762 = n^2 + n

0 = n^2 + n - 60672



Use the quadratic formula. I'll just use + instead of ±, since we can't have a negative number of terms.

n = [-b + √(b^2 - 4ac)] / 2a

n = [-1 + √(1^2 - 4(1)(-60672)] / 2(1)

n = [-1 + √(243049)] / 2

n = [-1 + 493] / 2

n = 492/2

n = 246 (Answer)



The first 246 terms of this arithmetic sequence, when added together, gives us 60762. Hope this helps!

Answer 2

1)

nth term = 48

sum = n/2[ a1+an]

546 = (26/2) [ a1+48]

546=13a1+624

13a1=-78

a1=-6



2) sum = n/2 [ 2a+(n-1)d]

781 = (22/2) [2a+(21)d]

781 = 11[2a+63]

781 =22a+(63)(11)

781 = 22a + 693

22a =88

a=4

first term is 4



3)

a+2d=9

a+6d=31

subtract

-4d =-22

d=11/2

a2=9-11/2=7/2 (second term)

a1= 7/2-11/2 =-4/2 =-2 (first term)



sum = n/2 [ 2a+(n-1)d]

n=22

sum = 11[2a+21d]

sum = 11[2(-2) + 21(11/2)]

= 11[-4+231/2]

= 11[223/2] =1226.5

Answer 3

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