Arithmetic sequence involving years

[adjacent]

Homework Statement

A 25 year old programme for building new houses began in Core Town in the year 1986 and finished in 2010.

The number of houses built each year form an arithmetic sequence. Given that 238 houses were built in 2000 and 108 in 2010, find the number of houses built in 1986

Homework Equations

$U_n = a + (n - 1)d$

The Attempt at a Solution

$d = \frac{108 - 238}{10} = -13$
$238 = a + ((2000- 1986) - 1) * -13$
$a = 407$

But the real answer is $a=420$ by adding 1 to the 2000-1968. I do not understand why.
Can someone enlighten me?

 

[rpthomps]

There's a mistake in your work above.

 

[adjacent]

There's a mistake in your work above.

Thanks. Fixed, but the question remains

 

[SammyS]

Homework Statement

A 25 year old programme for building new houses began in Core Town in the year 1986 and finished in 2010.

The number of houses built each year from an arithmetic equation. Given that 238 houses were built in 2000 and 108 in 2010, find the number of houses built in 1986

Homework Equations

$U_n = a + (n - 1)d$

The Attempt at a Solution

$d = \frac{108 - 238}{10} = -13$
$238 = a + ((2000- 1986) - 1) * -13$
$a = 407$

But the real answer is $a=420$ by adding 1 to the 2000-1968. I do not understand why.
Can someone enlighten me?

Do you mean that

The number of houses built each year form an arithmetic progression (sequence).​

?Define what is meant by n .

 

[adjacent]

Do you mean that

The number of houses built each year form an arithmetic progression (sequence).​

?Define what is meant by n .

Fixed. Thanks.
n is the number of years starting from 1986

 

[SammyS]

Fixed. Thanks.
n is the number of years starting from 1986

So, is 1986 year n=1 or is it year n=0 ?

 

[adjacent]

So, is 1986 year n=1 or is it year n=0 ?

n=1

 

[SammyS]

n=1

In that case, n = year - 1985

 

[adjacent]

In that case, n = year - 1985

Can you explain? I do not understand
I have exams in half an hour D:

 

[SammyS]

Can you explain? I do not understand
I have exams in half an hour D:

In other words, for the year 2000, n = 2000 -1985 = 15

 

[adjacent]

Thank god it did not come in exam

In other words, for the year 2000, n = 2000 -1985 = 15

I know but why 1985 instead of 1986? The question says 1986 is the starting year

 

[SammyS]

Thank god it did not come in examI know but why 1985 instead of 1986? The question says 1986 is the starting year

Well, as you said, for the year 1986 n should be 1. 1986 - 1986 = 0, not 1.

1987 should give n = 2, right? But 1987 - 1986 = 1 not 2. Etc.

 

[adjacent]

Well, as you said, for the year 1986 n should be 1. 1986 - 1986 = 0, not 1.

1987 should give n = 2, right? But 1987 - 1986 = 1 not 2. Etc.

I am so stupid. Thanks a lot. This was bugging me so hard last night.