In an AP, the 9th term is -4 times the 4th term and the sum of the 5th and 7th term is 9. Find the first term and the common difference

Answer 1

the 9th term is -4 times the 4th term ... a+8d = -4(a+3d)
the sum of the 5th and 7th term is 9 ... a+4d + a+6d = 9
Now just simplify those two equations, and solve for a and d.

Answer 2

"In an AP, the 9th term is -4 times the 4th term" ---> a+8d = -4(a+3d)
"he sum of the 5th and 7th term is 9" ---> a+4d + a+6d = 9

simplify each equation, the solve the system of 2 equations.

Answer 3

In AP:

a1 = first term

d = common difference

nth term = an

an = a1 + ( n - 1) d
________________________

a4 = a1 + ( 4 - 1 ) d = a1 + 3 d

a5 = a1 + ( 5 - 1 ) d = a1 + 4 d

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

a9 = a1 + ( 9 - 1 ) d = a1 + 8 d
________________________

Your conditions:

a9 = - 4 a4

a1 + 8 d = - 4 ( a1 + 3 d )

a1 + 8 d = - 4 a1 - 12 d

Add 4 a1 o both sides

5 a1 + 8 d = - 12 d

Subtract 8 d to both sides

5 a1 = - 20 d

Divide both sides by 5

a1 = - 4 d

a5 + a7 = 9

a1 + 4 d + a1 + 6 d = 9

2 a1 + 10 d = 9

Replace a1 = - 4 d in tis equation

2 ∙ ( - 4 d ) + 10 d = 9

- 8 d + 10 d = 9

2 d = 9

Divide both sides by 2

d = 9 / 2

d = 4.5

a1 = - 4 d

a1 = - 4 ∙ 9 / 2 = - 36 / 2 = - 18

First term a1 = - 18 , common difference d = 9 / 2 = 4.5

Your AP:

a1 = - 18

a2 = - 18 + 4.5 = - 13.5

a3 = - 13.5 + 4.5 = - 9

a4 = - 9 + 4.5 = - 4.5

a5 = - 4.5 + 4.5 = 0

a6 = 0 + 4.5 = 4.5

a7 = 4.5 + 4.5 = 9

a8 = 9 + 4.5 = 13.5

a9 = 13.5 + 4.5 = 18

Proof:

The 9th term is - 4 times the 4th term.

a9 = - 4 a 4

18 = - 4 ∙ ( - 4.5 )

18 = 18

The sum of the 5th and 7th term is 9.

a5 + a7 = 9

0 + 9 = 9

9 = 9

Answer 4

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