It is given that the sum of five consecutive terms of an arithmetic sequence is 5 and the product of these five consecutive terms is 45. Find the possible common differences of the sequence.

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Uniteasy Admin · Accepted Answer

$$ a_n= a_1+(n-1)d $$
$$ a_{n+1}= a_1+nd$$
$$ a_{n+2}=a_1+(n+1)d$$
$$ a_{n+3}=a_1+(n+2)d$$ 
$$ a_{n+4}=a_1+(n+3)d$$
$$S_5=\dfrac{5(a_n+a_{n+4})}{2}=\dfrac{5(5a_1+5nd+5d)}{2} $$
$$=\dfrac{25(a_1+nd+d)}{2} =5 $$
$$a_1+nd+d=\dfrac{2}{5} $$

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