In A.P the difference between the 8th term and 4th term is 20,and the 8th term is one and half times the 4th term.Find the common difference, and the first term.
term(8) - term(4) = 20
a + 7d - a - 3d = 20
4d = 20
d = 5
term(8) = (3/2)(term(4))
a+7d = (3/2)(a+3d)
2a + 14d = 3a + 9d
5d = a
a = 25
a=25 ---> first term
d = 5
check:
sequence is:
25, 30, 35, 40, 45, 50, 55, 60, ..
term(8) = 60, term(4) = 20
is 60-40 = 20 ? yes
is 60 = (3/2)(40) ? yes
Why did the term cross the sequence? To get to the seventh term! Ha-ha!
Okay, okay, let's solve this one. We know that the difference between the 8th and 4th terms is 20. So, let's say the common difference is "d" and the first term is "a."
The 8th term would then be a + 7d, and the 4th term would be a + 3d. And we know that the 8th term is 1.5 times the 4th term.
So, we can write the equation as:
a + 7d = 1.5(a + 3d)
Now, let's do some algebraic acrobatics to solve this!
Expand the equation:
a + 7d = 1.5a + 4.5d
Combine like terms:
6.5d = 0.5a
Divide both sides by 0.5:
13d = a
So, the first term is 13d and the common difference is, well, "d" itself! Ha-ha!
Hope that clears things up!
To find the common difference and the first term in an arithmetic progression (A.P.), you need to solve the given information using the general formulas of A.P.
Let's assume that the first term of the A.P. is 'a' and the common difference is 'd.'
Given information:
The difference between the 8th term and the 4th term is 20, which means that the 8th term (denoted by 'a8') minus the 4th term (denoted by 'a4') is 20:
a8 - a4 = 20 (Equation 1).
The 8th term is one and a half times the 4th term, which can be written as:
a8 = 1.5 * a4 (Equation 2).
Now, let's solve the equations to find the common difference ('d') and the first term ('a').
Step 1: Substitute Equation 2 into Equation 1.
1.5 * a4 - a4 = 20
0.5 * a4 = 20
a4 = 20 / 0.5
a4 = 40
Step 2: Substitute the value of a4 into Equation 2 to find a8.
a8 = 1.5 * 40
a8 = 60
Step 3: Substitute the values of a4 (40) and a8 (60) into the formula for the nth term of an A.P., which is given by:
an = a + (n-1)d
Using the values:
60 = a + (8-1)d
40 = a + (4-1)d
Step 4: Subtract the second equation from the first to eliminate 'a.'
20 = 3d
Step 5: Divide both sides of the equation by 3 to solve for 'd.'
d = 20 / 3
d ≈ 6.67 (rounded approximately)
Step 6: Substitute the value of 'd' back into the second equation to find 'a.'
40 = a + (4-1)(6.67)
40 = a + 3(6.67)
40 = a + 20.01
a = 40 - 20.01
a ≈ 19.99 (rounded approximately)
Therefore, the common difference (d) ≈ 6.67 and the first term (a) ≈ 19.99 (rounded approximately).
To find the common difference and the first term in an arithmetic progression (A.P.), we can use the given information about the terms.
Let's assume that the first term of the A.P. is 'a' and the common difference is 'd'.
We are given two pieces of information:
1. The difference between the 8th term and the 4th term is 20: 8th term - 4th term = 20
2. The 8th term is one and a half times the 4th term: 8th term = 1.5 * 4th term
To get the first term ('a'), we need to find the value of the 4th term.
Using the second condition, we have: 8th term = 1.5 * 4th term
Substituting the values in terms of 'a' and 'd', the equation becomes: a + 7d = 1.5 * (a + 3d)
Expanding the equation, we get: a + 7d = 1.5a + 4.5d
To find the common difference ('d'), we can solve these two equations simultaneously.
1. 8th term - 4th term = 20
(a + 7d) - (a + 3d) = 20
4d = 20
d = 5
Now that we know the common difference ('d') is 5, we can substitute it into any of the earlier equations to find the value of the first term ('a').
Using equation a + 7d = 1.5a + 4.5d, we have:
a + 7(5) = 1.5a + 4.5(5)
a + 35 = 1.5a + 22.5
a - 1.5a = 22.5 - 35
-0.5a = -12.5
a = (-12.5)/(-0.5)
a = 25
Therefore, the common difference (d) is 5, and the first term (a) is 25.