# Arithmetic sequence involving years

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## 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:

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
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Homework Helper
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## 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
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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 ?

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

SammyS
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Homework Helper
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n=1
In that case, n = year - 1985

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

SammyS
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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

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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
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Homework Helper
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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.