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Question: Complete the pattern: \(8,3, - 2, - 7,\\_,\\_\)....

Complete the pattern: 8,3, - 2, - 7,\\_,\\_.

Explanation

Solution

In order to find the next term of the series, we should first know that which kind of series is being followed, whether it is in Arithmetic Progression or Geometric progression, or any other series. In order to check that, we would subtract the first term from second and second from third, if both give the same results, then it’s in Arithmetic Progression (AP), if not then we will check for Geometric Progression and others.

Complete step by step answer:
We are given with the series 8,3, - 2, - 7,\\_,\\_. Where the first term is T1=8{T_1} = 8, second term is T2=3{T_2} = 3, third term is T3=−2{T_3} = - 2. To check for Arithmetic Progression, subtracting first term from second and we get:
T2−T1=3−8=−5{T_2} - {T_1} = 3 - 8 = - 5
Subtracting second term from third term, and we get:
T3−T2=−2−3=−5{T_3} - {T_2} = - 2 - 3 = - 5

Since, the difference is the same, so the given series is an AP, and the common difference is d=−5d = - 5. The next term in an AP can be found by adding the common difference to the previous number.Similarly, in the given series the last term known is −7 - 7.

To find its next term adding the common difference d=−5d = - 5 to −7 - 7, and we get: −7+d=−7+(−5)=−7−5=−12 - 7 + d = - 7 + \left( { - 5} \right) = - 7 - 5 = - 12, which is the term.
Similarly, for another term adding common difference d=−5d = - 5 to its previous term that is −12 - 12 and we get: −12+d=−12+(−5)=−12−5=−17 - 12 + d = - 12 + \left( { - 5} \right) = - 12 - 5 = - 17

Therefore, the complete series obtained is: 8,3,−2,−7,−12,−178,3, - 2, - 7, - 12, - 17.

Note: Since, we got to know that the series was in AP, if it was not then we would have checked for GP, and for that we would have divided the second term by first and third term by second, if those two give the same results, then it’s in GP. If not GP, then we will check for another series.