KACL-DS-INC

--> Knowing that the 7th term of an arithmetical progression is

--> 11 plus (more) than 4 times the 4 term of the A.P.

--> What is the 2 th term of the series if the sum of the 15 terms.

--> of the A.P is 19.

--> Calculate the sum of the 27 terms of the series.

--> Produced by MATCAL Program

--> The Nth term in an A.P is given by the equation below

--> N(th) = [ a + ( n - 1 ) x d ] , Incremental value = 4

--> N = The nth term to be found, a = first term, d = common

difference

--> 7th = a + ( 7 - 1 ) x d

--> 7th = a + ( 6 ) x d

--> 7th = a + 6d

--> 4th of the A.P = a + 3d

--> 4th = a + ( 4 - 1 ) x d

--> 4th = a + ( 3 ) x d

--> 4th = a + 3d

--> 4th of the A.P = a + 3d

--> 7th = a + 6d = 4th = a + 3d

--> a + 6d = 4 x ( a + 3d ) + 11

--> a + 6d = 4a + 12d + 11

--> Combine like terms together

--> -3a + -6d = 11

--> -3a + -6d = 11 --> Eq(1)

--> Sum of the terms = 19 , and the N(th) term = 15

--> Sum of the terms = S(n) = n / 2 [2a + (n-1) x d]

--> 19 = 15 / 2 [2a + (15 -1) x d]

--> 19 = 7.5 x [2a + (15 -1) x d]

--> 19 = 7.5 x [2a + ( 14 ) x d]

--> 19 = 7.5 x [ 2a + 14d ]

--> 19 = 15a + 105d

--> 19 = 15a + 105d --> Eq(2)

--------------------------------------->

--> -3a + -6d = 11 --> Eq(1)

--> 15a + 105d = 19 --> Eq(2)

--------------------------------------->

--> Solving by matrices

--------------------------------------->

--> Solving for the value of the determinant =

--> [ -3 -6 | 11 ]

--> [ 15 105 | 19 ]

--------------------------------------->

--> Det = [ -3 x 105 - 15 x -6 ]

--> Det = -3 x 105 - 15 x -6

--> Det = -225

--> Det = -225.00

--> Produced by MATCAL Program

--> Solving for the value of a =

--> [ 11 -6 ]

--> [ 19 105 ]

--> a = [ 11 x 105 - 19 x -6 ]

--> a = 1269

--> a = 1269 / -225

--> a = -5.64

--> Produced by MATCAL Program

--> Solving for the value of d =

01-09- Everything.mp3

--> [ -3 11 ]

--> [ 15 19 ]

--> d = [ -3 x 19 - 15 x 11 ]

--> d = -222

--> d = -222 / -225

--> d = 0.986666666666667

--> Produced by MATCAL Program

--> a = -5.64

--> d = 0.99

--> Produced by MATCAL Program

--> Checking for error and accuracy

--> 15a + 105d = 19--> Eq(2)

--> 15 x -5.64 + 105 x 0.986666666666667 = 19 --> Eq(2)

--> -84.6 + 103.6 = 19 --> Eq(2)

--> 19 = 19 --> Eq(2)

--> Produced by MATCAL Program

--> The 2th term = a + (n - 1) x d

--> The 2th term = a + ( 2 - 1 ) x d

--> The 2th term = a + 1d

--> a = -5.64

--> d = 0.99

--> The 2th term = -5.64 + 1d

--> The 2th term = -5.64 + 0.986666666666667

--> The 2th term = -4.65333333333333

--> The 2th term = -4.65

--> The 2th term of the series (A.P) = -4.65

--> Produced by MATCAL Program

--> S(n) = n / 2 [2a + (n-1) x d]

--> n = 27 , a = -5.64 , d = 0.99

--> S(27) = 27 / 2 [ 2 x -5.64 + ( 27 - 1 ) x 0.99 ]

--> S(27) = 13.5 x [ -11.28 + 26 ) x 0.99 ]

--> S(27) = 13.5 x [ -11.28 + ( 25.74 ]

--> S(27) = 13.5 x [ 14.3733333333333 ]

--> S(27) = 194.04

--> The sum of the 27 terms of the A.P = 194.04

--> Produced by MATCAL Program

--> Using simultaneous equation to solve for a and d

--> -3a + -6d = 11 --> Eq(1)

--> 15a + 105d = 19 --> Eq(2)

--> Solve for d in EQ(2) using elimination method

--> Multiply EQ(1) by 15 / -3 add it to EQ(2) below to

eliminate (a)

--> 45a / -3 + -15 x -6d = 11 x -15 --> Eq(3)

--> -15a + -30d = 55 --> Eq(3)

--> Add EQ(3) to EQ(2) to eliminate (a) and solve for (d)

--> Right side is equal to = 55 + 19

--> Right side is equal to = 74

--> -15a + 15a = 0 , 105d + -30d = 75d = 74

--> 75d = 74

--> d = 74 / 75

--> d = 0.986666666666667

--> d = 0.99

--> Solve for a by substituting d into EQ(2) above

--> 15a + 105d = 19 --> Eq(2)

--> a = ( 19 - 105d ) / 15 --> Eq(4)

--> a = ( 19 - 105 x 0.986666666666667 ) / 15 --> Eq(4)

--> a = ( 19 - 103.6 ) / 15 --> Eq(4)

--> a = ( -84.6 ) / 15 --> Eq(4)

--> a = -5.64

--------------------------------------->

--> a = -5.64

--> d = 0.99

--------------------------------------->

--> Produced by MATCAL Program

--> It is clearly obvious that the value of a and d are correct

--> because the answer produced by the matrices and

simultaneous equation are the same.

--> Your answers are coreect and you score a point.

--> Click on next or press F7 key to move to the next screen.

--> Produced by MATCAL Program

15 - In The Air Tonight.mp3