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
p;2p+2;5p+3 arithmetic sequence determine the next three terms
3 answers
find p so that
2p+2 - p = (5p+3)-(2p+2)
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
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