# Solving Arithmetic Progression

#### arangu1508

I have enclosed the problem encountered by me today. The answer given is 2. I searched in internet. There are solutions which I am not able to follow. I understand common difference should be a fraction but I feel the common difference me a negative fraction. I am puzzled. I am not able to follow further. Guidance required please.

With thanks and regards
Aranga

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MHF Helper

#### arangu1508

(11/2)(2a+10d)=33
2a+10d=6
a+5d=3
a+3d = ?
a is a positive fraction. What it signifies frankly I don't know.
Alternative number is an integer.
So I inferred 4th term is an integer.
How to collate all these information?

#### Debsta

MHF Helper
This is either a badly worded question or one that has multiple answers.
If you assume that d=0.5, then every alternate term will be an integer as long as the first term is an integer + 0.5.
But this will also happen if d= any integer + 0.5.
For now, assume d=0.5, can you find a?

arangu1508

#### Debsta

MHF Helper
Maybe the question needs to say the first term is a positive PROPER fraction?

#### arangu1508

They have given choices for the answers.
2, 3, 5 and 6.
Out of which they have given the answer as 2.
Whether we have to go from the choices?
In internet they have given the explanation. I was not able to follow.

#### Idea

$$\displaystyle \left\{\frac{21}{2},9,\frac{15}{2},6,\frac{9}{2},3,\frac{3}{2},0,-\frac{3}{2},-3,-\frac{9}{2}\right\}$$

is an example where the fourth term = 6

So 6 is possible, so is 2

#### Debsta

MHF Helper
If the answer they want is 2, then the question should have said that the first term is a PROPER fraction. Then Idea's answer of 6 would be invalid since 21/2 is an improper fraction. As the question is written though, both 2 and 6 are possible answers.

#### arangu1508

thank you very much. The discussions were very useful.