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Arithmetical Progression

```
Date: 7/7/96 at 1:55:28
From: Wong Cheong Siong
Subject: Arithmetical Progression

Dear Dr. Math,

An arithmetical progression has a common difference of 1/1/2. The
sum of the first n terms is 365 and the sum of the first 2n terms is
1330. Calculate the value of n and the first term.

Best Regards,
Wong Cheong Siong
```

```
Date: 7/7/96 at 16:34:41
From: Doctor Anthony
Subject: Re: Arithmetical Progression

a = first term, d= 1.5 (I think, because you wrote it 1/1/2 which
could equal 2)  Sn = (n/2)(2a+(n-1)d)

First situation (n/2)(2a + (n-1)*1.5) = 365

Second situation (2n/2)(2a + (2n-1)*1.5) = 1330

These can be written  n(2a + 1.5(n-1)) = 730
n(2a + 1.5(2n-1)) = 1330

Subtract top equation from the second equation.  This elimimates 2an
term.

We get     1.5n(2n-1 - n+1) = 600
1.5n(n) = 600
n^2 = 400
n = 20

To get first term we can write  20(2a + 1.5*19) = 730
4a + 3*19 = 73
4a = 16
a = 4

Check if this works in second situation.

20(8 + 1.5*39) = 1330 which checks.

-Doctor Anthony,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```

```
Date: 7/9/96 at 3:45:12
From: Wong Cheong Siong
Subject: Re: Arithmetical Progression

Dear Dr. Math,
I would like to thank you for answering my question. However, my
teacher told us that a question can be done in very few steps, but he
will not tell us how. So, can I ask you to show me the fastest
possible way to solve this question:

The sum of n terms of a certain series is 3n^2 +  10n for all values
of n. Find the nth term and show that the series is an arithmetical
progression.

Best Regards,
Wong Cheong Siong
```

```
Date: 7/9/96 at 7:28:55
From: Doctor Anthony
Subject: Re: Arithmetical Progression

To show whether it is an AP or a GP you can find the first three terms
using the formula for Sn = 3n^2 + 10n

S1 = 3 + 10 = 13        a1 = 13
S2 = 12 + 20 = 32       a2 = S2-S1 = 19
S3 = 27 + 30 = 57       a3 = S3-S2 = 25

The series has a = 13  and there is a common difference  d = 6

The nth term is given by  a + (n-1)d

= 13 + (n-1)*6
= 6n + 7

-Doctor Anthony,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Sequences, Series

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