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
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.
"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.
In AP:
a1 = first term
d = common difference
nth term = an
an = a1 + ( n - 1) d
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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
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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
Continue doing this you are helping us more and understanding.
To solve this problem, we need to use the properties of an arithmetic progression (AP). An AP is a sequence in which the difference between any two consecutive terms is constant.
Let's assume the first term of the AP is 'a', and the common difference is 'd'.
We are given two pieces of information:
1. The 9th term is -4 times the 4th term:
The 4th term can be written as: a + 3d (since there are 3 terms between the 4th and 9th terms).
Hence, the 9th term can be written as: a + 8d.
According to the given information, we have the equation: a + 8d = -4(a + 3d).
2. The sum of the 5th and 7th term is 9:
The 5th term can be written as: a + 4d.
The 7th term can be written as: a + 6d.
According to the given information, we have the equation: (a + 4d) + (a + 6d) = 9.
Now, we have two equations with two variables. We can solve them simultaneously to find the values of 'a' and 'd'.
Equation 1: a + 8d = -4(a + 3d)
Simplifying, we get: a + 8d = -4a - 12d
Rearranging terms, we have: 5a = -20d
Dividing both sides by 5, we get: a = -4d. --- Equation 3
Equation 2: (a + 4d) + (a + 6d) = 9
Simplifying, we get: 2a + 10d = 9.
Substituting a = -4d from Equation 3, we get: 2(-4d) + 10d = 9
Simplifying further, we have: -8d + 10d = 9
Combining like terms, we get: 2d = 9
Dividing both sides by 2, we get: d = 4.5.
Now that we have the value of 'd', we can find the value of 'a' using Equation 3:
a = -4d = -4(4.5) = -18.
Therefore, the first term of the AP is -18, and the common difference is 4.5.