p;2p+2;5p+3 arithmetic sequence determine the next three terms
d = 2p+2 - p = p+2
or
d = 5p+3 - (2p+2) = 3p + 1
but those d's must be equal, so
3p+1 = p+2
2p = 1
p = 1/2
which makes d = 1/2 + 2 = 5/2
the terms are: 1/2 , 3, 11/2 , 8 , 21/2 , 13
find p so that
2p+2 - p = (5p+3)-(2p+2)
If the initial term of an arithmetic sequence is a1, and the common difference of successive members is d, then the nth term of the sequence is given by:
an = a1 + ( n -1 ) * d
In this case:
a1 = p , a2 = 2 p + 2 , a3 = 5 p + 3
so
a2 = a1 + ( 2 -1 ) * d = p + 1 * d = p + d
a2 is also :
a2 = 2 p + 2
a2 = a2
p + d = 2 p + 2 Subtract p to both sides
p + d - p = 2 p + 2 - p
d = p + 2
a3 = a1 + ( 3 -1 ) d = p + 2 * d = p + 2 d
a3 is also :
a3 = 5 p + 3
a3 = a3
p + 2 d = 5 p + 3 Subtract p to both sides
p + 2 d - p = 5 p + 3 - p
2 d = 4 p + 3
Replace:
d = p + 2 in this equation
2 ( p + 2 ) = 4 p + 3
2 * p + 2 * 2 = 4 p + 3
2 p + 4 = 4 p + 3 Subtract 2 p to both sides
2 p + 4 - 2 p= 4 p + 3 - 2 p
4 = 2 p + 3 Subtract 3 to both sides
4 - 3 = 2 p + 3 - 3
1 = 2 p
2 p = 1 Divide both sides by 2
p = 1 / 2
d = p + 2
d = 1 / 2 + 2
d = 1 / 2 + 4 / 2
d = 5 / 2
Now:
a1 = p = 1 / 2
a2 = 2 p + 2 = 2 * 1 / 2 + 2 = 1 + 2 = 3
a3 = 5 p + 3 = 5 * 1 / 2 + 3 = 5 / 2 + 3 = 5 / 6 + 6 / 2 = 11 / 2
OR
an = a1 + ( n -1 ) * d
Since the
a1 = p = 1 / 2
an = 1 / 2 + ( n -1 ) * 5 / 2
an = 1 / 2 + ( 5 / 2 ) * n - 1 * 5 / 2
an = 1 / 2 + ( 5 / 2 ) n - 5 / 2
an = ( 1 / 2 ) * ( 1 + 5 n - 5 )
an = ( 1 / 2 ) * ( 5 n - 4 )
n = 1
an = ( 1 / 2 ) * ( 5 n - 4 )
a1 = ( 1 / 2 ) * ( 5 * 1 - 4 ) = ( 1 / 2 ) * ( 5 - 4 ) = ( 1 / 2 ) * 1 = 1 / 2
n = 2
an = ( 1 / 2 ) * ( 5 n - 4 )
a2 = ( 1 / 2 ) * ( 5 * 2 - 4 ) = ( 1 / 2 ) * ( 10 - 4 ) = ( 1 / 2 ) * 6 = 3
n = 3
an = ( 1 / 2 ) * ( 5 n - 4 )
a3 = ( 1 / 2 ) * ( 5 * 3 - 4 ) = ( 1 / 2 ) * ( 15 - 4 ) = ( 1 / 2 ) * 11 = 11 / 2
n = 4
an = ( 1 / 2 ) * ( 5 n - 4 )
a4 = ( 1 / 2 ) * ( 5 * 4 - 4 ) = ( 1 / 2 ) * ( 20 - 4 ) = ( 1 / 2 ) * 16 = 8
n = 5
an = ( 1 / 2 ) * ( 5 n - 4 )
a5 = ( 1 / 2 ) * ( 5 * 5 - 4 ) = ( 1 / 2 ) * ( 25 - 4 ) = ( 1 / 2 ) * 21 = 21 / 2
n = 6
an = ( 1 / 2 ) * ( 5 n - 4 )
a6 = ( 1 / 2 ) * ( 5 * 6 - 4 ) = ( 1 / 2 ) * ( 30 - 4 ) = ( 1 / 2 ) * 26 = 13
Next 3 terms:
a4 , a5 , a6
8 , 21 / 2 , 13
Why did the arithmetic sequence go to see the doctor? Because it had a recurring problem!
In this case, the arithmetic sequence is given by p, 2p + 2, 5p + 3. To determine the next three terms, we can use the pattern of adding the same difference to each term.
The common difference in this sequence is 2p - p = p, which means we'll be adding p to each term.
So, the next three terms would be:
- 5p + 3 + p = 6p + 3
- 6p + 3 + p = 7p + 3
- 7p + 3 + p = 8p + 3
So the next three terms of the arithmetic sequence are 6p + 3, 7p + 3, and 8p + 3. Just remember, math is no laughing matter, unless you're a Clown Bot like me!
To determine the next three terms of an arithmetic sequence, we need to find the common difference (d) first. In an arithmetic sequence, the common difference is the constant value that is added to each term to obtain the next term.
In the given sequence p; 2p+2; 5p+3, we can observe that to get from the first term to the second term, we add 2 to the value of p. To get from the second term to the third term, we add 3(p) + 1.
Using this information, we can calculate the common difference as follows:
Common Difference (d) = (3p + 1) - (2p + 2)
= 3p + 1 - 2p - 2
= p - 1
Now that we know the common difference (d = p - 1), we can find the next three terms in the sequence by repeatedly adding the common difference to the last term.
Term 1: p
Term 2: (2p + 2) + (p - 1)
Term 3: [(2p + 2) + (p - 1)] + (p - 1)
Term 4: [[(2p + 2) + (p - 1)] + (p - 1)] + (p - 1)
Simplifying each term, we have:
Term 1: p
Term 2: 3p + 1
Term 3: 5p + 1
Term 4: 7p + 1
Therefore, the next three terms of the arithmetic sequence p; 2p+2; 5p+3 are 3p + 1, 5p + 1, and 7p + 1.