find the value of p so that p+5, 4p+3, and 8p-2 will from successive terms of an arithmetic sequence, then find the three successive terms.
p+5, 4p+3, 8p-2
If they are in arithmetic sequence, then have a common difference or
(4p + 3) - (p + 5) = (8p - 2) - (4p + 3)
Solving for p,
4p + 3 - p - 5 = 8p - 2 - 4p - 3
3p - 2 = 4p - 5
3p - 4p = -5 + 2
-p = -3
p = 3
To get the value of each term just substitute the value of p in each term.
hope this helps~ `u`
as we formula:2b=a+c
2(4p+3)=p+5+8p-2
8p+6=9p+3
-p=-3
p=3
1)8
2)15
3)22
To find the value of p that makes the three given expressions form successive terms of an arithmetic sequence, we need to determine if there is a common difference between these terms.
In an arithmetic sequence, the common difference (d) is constant, which means the second term minus the first term should be equal to the third term minus the second term.
So, we can set up the following equation:
(4p + 3) - (p + 5) = (8p - 2) - (4p + 3)
Simplifying the equation:
3p - 2 = 4p - 5
Rearranging the equation:
4p - 3p = 5 - 2
p = 3
Thus, the value of p that makes p + 5, 4p + 3, and 8p - 2 form successive terms of an arithmetic sequence is p = 3.
To find the three successive terms, substitute the value of p = 3 into the expressions:
First term: p + 5 = 3 + 5 = 8
Second term: 4p + 3 = 4(3) + 3 = 12 + 3 = 15
Third term: 8p - 2 = 8(3) - 2 = 24 - 2 = 22
Therefore, the three successive terms are 8, 15, and 22.
To find the value of "p" and the three successive terms, we need to solve two equations.
First, let's write the equation for an arithmetic sequence. In an arithmetic sequence, the difference between any two consecutive terms is constant. Let's call that common difference 'd'.
So, the second term in the sequence is obtained by adding 'd' to the first term, while the third term is obtained by adding 'd' to the second term.
1st term: p + 5
2nd term: (p + 5) + d
3rd term: [(p + 5) + d] + d
Now, we can set up the equation using the given information:
(p + 5) + d = 4p + 3
[(p + 5) + d] + d = 8p - 2
Let's solve these equations step by step.
1) (p + 5) + d = 4p + 3
First, simplify by removing parentheses:
p + 5 + d = 4p + 3
Next, group the variables:
p - 4p = 3 - 5 - d
Simplify:
-3p = -2 - d
Now, isolate 'p' by dividing both sides by -3:
p = (2 + d) / 3
2) [(p + 5) + d] + d = 8p - 2
Start by removing the parentheses:
p + 5 + d + d = 8p - 2
Combine like terms:
p + 5 + 2d = 8p - 2
Now, group the variables:
p - 8p = -2 - 5 - 2d
Simplify:
-7p = -7 - 2d
Divide by -7 to isolate 'p':
p = (-7 - 2d) / -7
Since we have two expressions for 'p' from the two equations, we can equate them to find the value of 'd':
(2 + d) / 3 = (-7 - 2d) / -7
To solve this equation, cross-multiply:
(-7 - 2d) * 3 = (2 + d) * -7
Expand and simplify:
-21 - 6d = -14 - 7d
Add 7d and subtract 14 from both sides:
d = -7
Now that we have the value of 'd', we can substitute it back into one of the equations for 'p':
p = (2 + (-7)) / 3 = -1
So, the value of 'p' is -1, and the three successive terms are:
1st term: p + 5 = -1 + 5 = 4
2nd term: (p + 5) + d = 4 - 7 = -3
3rd term: [(p + 5) + d] + d = -3 - 7 = -10
Therefore, the three successive terms of the arithmetic sequence are 4, -3, and -10.