Question

p;2p+2;5p+3 arithmetic sequence determine the next three terms

Answer 1

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

Answer 2

find p so that

2p+2 - p = (5p+3)-(2p+2)

Answer 3

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