Showing results for The sum off the first three term of an arithmetic sequence is 219 and the sum of the first nine terms in the same sequence is 603. find the

Question

Showing results for The sum off the first three term of an arithmetic sequence is 219 and the sum of the first nine terms in the same sequence is 603. find the fifth term.

Answer 1

S(3)=219=(3/2)(2a+[3-1]d)

S(9)=603=(9/2)(2a+[9-1]d)

219=(3/2)(2a+2d)

603=(9/2)(2a+8d)

219=3a+3d

603=9a+36d

simply both equations

73=a+d

67=a+4d

subtract

6=-3d

d=-2

73=a+(-2)

a=75

use simple arithmetic or the formula

75, 73, 71, 69, 67 (answer), 65, 63, 61, 59

65 is the answer

x(5)=75+(-2)(5-1)

x(5)=75+(-2)(4)

x(5)=75+(-8)

x(5)=67

 

 

 

Answer 2

Hi Suama,

 

 

  The sum off the first three term of an arithmetic sequence is 219 and the sum of the first nine terms in the same sequence is 603. find the fifth term.

 

   The first term a₁

   The second term is a₁ + d   where d is the common difference between two terms

   The third term is    a₁ +d +d =a₁ +2d

 

sum off the first three term of an arithmetic sequence is 219:

 

   a₁ + a₁ +d + a₁ +2d = 219

                    3a₁ +2d  = 219

 

the sum of the first nine terms in the same sequence is 603

 

If you follow the same pattern as above you will get:   9a₁ + 8d=603

 

Now there is a system of equations:

 

                          3a₁ +2d = 219

                          9a₁ +8d = 603

 

Multiply the first equation by -3 and then add:   

 

            -3[3a₁ +2d = 219]       -9a₁ + -6d = -657

                9a₁ +8d =603           9a₁ + 8d  = 603

                                                -------------------

                                                   0  + 2d  = -54

                                                            d  =  -27

 

Substitute back into first equation:  3a1 +2d = 219

                                                    3a₁ +2(-27)=219

                                                    3a₁ - 54 = 219

                                                            3a₁ =273

                                                             a₁ = 91

 

 

Find the fifth term.

  The first term is 91. The common difference d= -27.

 

   The fifth term is     a₅= a₁ +4d

                               a₅= 91 +4(-27)

                               a₅=91-108

                               a₅=17

 

 

  ANSWER:  The fifth term is 17.

Hope that helps you understand the process.

THINK MATH!