Complete the following series 3, 7, 13, 21, 31………….. a) 40 b) 41 c) 42 d) 43
Hint: Observe the successive difference of the given sequence. Try to write a sequence of the difference and observe them to get the next number in the given sequence, try it.
Complete step-by-step answer: The sequence in the problem is 3, 7, 13, 21, 31………… Here we need to determine the term after 31 in the sequence, so we need to observe the given sequence and try to get some symmetric behaviour in the given sequence. Now, observe the difference between the successive terms, as we can notice that the difference between second and first term is 7 – 3 = 4 and difference between third term and second term is 6. Similarly the difference between fourth and third term is ‘8’ and hence similarly the difference between 5th term and 4th term is 10. So, we can notice that the difference between the successive terms are 4, 6, 8, 10. It means the successive differences in this sequence is 2. It means the successive difference of the given sequence is increasing by the value of 2. So, the next term in the sequence of successive differences would be 4, 6, 8, 10, 12, 14,………. So we can get the next term by adding ‘12’ to the previous term i.e. 31. Hence we get the next term as 31 + 12 = 43. So, the sequence can be given as now; 3, 7, 13, 21, 31, 43. So, option (d) is correct.
Note: One may go wrong if he or she tries to relate the given question from the series of A.P or G.P. As the given sequence is not an A.P or G.P, but their differences are in A.P with the common difference 2. Observing the successive difference of the terms is the key point of the sequence and the question. Don’t go for calculating the general term of the sequence. As we do not need to calculate the higher terms of the sequence. So, just observe and get the 6th term easily without using any formula or identity.