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Question: Look at the series: \(544,509,474,439,...\) What number should come next?...

Look at the series: 544,509,474,439,...544,509,474,439,...
What number should come next?

Explanation

Solution

The given question requires us to find the next term in the series whose first few terms are provided to us. So, we find a general formula for the terms of the sequence given to us. Hence, we have to generalize a formula for the terms of the given sequence. We have to find out whether the given series or sequence is an arithmetic progression, a geometric progression, a harmonic progression, an arithmetic geometric progression or a special type of series.

Complete step by step answer:
The given problem puts our analytical skills to test. We have to first identify the nature of the given sequence or series and then find a generalized formula for the terms of the sequence.
The sequence given to us is: 544,509,474,439,...544,509,474,439,....
First checking the given series for arithmetic progression.
So, we calculate the difference between consecutive terms.
The difference between the first two terms of the sequence is 509−544=−35509 - 544 = - 35.
Similarly, the difference between the next two terms of the sequence is 474−509=−35474 - 509 = - 35.
Similarly, the difference between any two consecutive terms of an arithmetic progression is −35 - 35.
Since the difference between the consecutive terms of the series is constant, hence the series is an arithmetic progression.
Here, the first term of AP is a=544a = 544 and the common difference is d=−35d = - 35.
Now, we have to calculate the next term in the series after 439439. So, we have to calculate the 5th{5^{th}} term in the AP. So, we know the formula of the general nth term of an AP is equal to an=a+(n−1)d{a_n} = a + \left( {n - 1} \right)d.
Hence, we get the fifth term as a5=a+(5−1)d=a+4d{a_5} = a + \left( {5 - 1} \right)d = a + 4d. Hence, substituting the values of a and d in the formula, we get,
a5=544+4(−35){a_5} = 544 + 4\left( { - 35} \right)
Simplifying the calculations, we get,
⇒a5=544−140=404\Rightarrow {a_5} = 544 - 140 = 404

Note:
In such a type of question, we should first find out the nature of the series and then try to figure out the general term of the series. In this way, we would have an idea beforehand of what the formula for the general term of the sequence would look like. One must take care of the calculations in order to be sure of the final answer.