# Arithmetic sequence involving years

Gold Member

## 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?

Last edited:

Related Precalculus Mathematics Homework Help News on Phys.org
There's a mistake in your work above.

Gold Member
There's a mistake in your work above.
Thanks. Fixed, but the question remains

SammyS
Staff Emeritus
Homework Helper
Gold Member

## 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 .

Gold Member
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
Staff Emeritus
Homework Helper
Gold Member
Fixed. Thanks.
n is the number of years starting from 1986
So, is 1986 year n=1 or is it year n=0 ?

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

SammyS
Staff Emeritus
Homework Helper
Gold Member
n=1
In that case, n = year - 1985

Gold Member
In that case, n = year - 1985
Can you explain? I do not understand
I have exams in half an hour D:

SammyS
Staff Emeritus
Homework Helper
Gold Member
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

Gold Member
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
Staff Emeritus
Homework Helper
Gold Member
Thank god it did not come in exam

I 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.

Gold Member
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.