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The question is: the first three terms of an arithmetic series $c_{n}$ are $$a(1+b), a(1+3b),a(1+5b)$$ I needed to find the common difference in terms of $a$ and $b$ and then find the expression for $c_{n}$.

The final part I struggled with where I have to find $a$ and $b$ and the information given is $$c_{5} = 25,c_{10} = 55$$

The answers for the first two parts are $difference = 2ab$ and $c_{n}=a(1+(2n-1)b)$

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The difference between the fifth term and the tenth term is five times the common difference, or $10ab$. Hence $10ab = 55-25 = 30$, so $ab=3$.

The fifth term is also $a(1+9b) = a+9ab = a+27 = 25$, so $a = -2$, which means $b = 3/a = -3/2$.

Thus $(a,b) = (-2, -3/2)$.

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On simplifying , the first three terms of series become a+ab , a+3ab , a+5ab .Therefore the common difference of the series is 2ab.

Now the nth term of the series is: (a+ab)+(n-1)(2ab).

on simplifying it becomes

a(1+(2n-1)b)

since the 5th and 10th term of series are 25 and 55.we get the relations:

25=a(1+9b) ... (1)

55=a(1+19b) ... (2)

dividing equation (1) by equation (2) , we get the value of b=-3/2 and substituting it in equation 1 we get the value of a = -2.

If there is any doubt,feel free to ask in comments.

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