- This time, you will be finding the first 4 terms of the sequence with a general term of an = 3n - 2 + n2
- For our second problem, we have to find the first 3 terms of the sequence with a general term of f(n) = n2 + 2n - 5.
- By substituting the 1, 2, and 3 to n, we can solve the first 3 terms of the given sequence. So we have, a1 = f(1) = (1)2 + 2(1) - 5 = -2 a2 = f(2) = (2)2 + 2(2) - 5 = 3a3 = f(3) = (3)2 + 2(3) - 5 = 10
- Did you know that we also have a type of sequence wherein the first few terms of the sequence are given (the first few terms that are given are called initial conditions) and succeeding terms are generated from preceding terms by solving a recurrence equation? This is what we call a recursive sequence or recurrence sequence. The famous Fibonacci sequence is a special type of recursive sequence.
- Finding the next 3 terms of the recursive sequence with an equation of an = an-1 + 5, a1 = 7 for n 1
- Let's take a look at some examples of recursive sequences.
- For the Fibonacci sequence, it always starts with 1 and 1. So the next terms would be 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8, 5 + 8 = 13, and so on. This sequence is named after Leonardo Fibonacci.
- In the case of a recursive sequence, we need to find the term after the initial condition. For this instance, we need to find the 2nd term. To find the 2nd term, we need to use 2 as the value of our n.
- Given: an = an-1 + 5, a1 = 7a2 = a2-1 + 5 = a1 + 5 = 7 + 5 = 12a3 = a3-1 + 5 = a2 + 5 = 12 + 5 = 17a4 = a4-1 + 5 = a3 + 5 = 17 + 5 = 22
- This time, let's discuss the series. To tell you honestly, sequence and series come in pairs because the series is just the sum of terms of the sequence.
- Since a sequence can be finite or infinite, this also holds true to a series, since it is dependent on the nature of the sequence.
- A partial sum is the sum of a part of a sequence. The sum of the first n terms of an infinite sequence is the nth partial sum.
- So based on the definition, S1 = a1 S2 = a1 + a2 or S2 = S1 + a2S3 = a1 + a2 + a3 or S3 = S2 + a3...Sn = a1 + a2 + ... + an
- For example, finding the 4th partial sum of the sequence generated by a general term an = 2n + 3
- At this point, try this one. Find the 6th partial sum of the sequence generated by the general term an = n3 - 2n + 11.
- To answer this problem, you must identify first the first 4 terms of the sequence. So by using substitution, General term : an = 2n + 3a1 = 2(1) + 3 = 5a2 = 2(2) + 3 = 7a3 = 2(3) + 3 = 9a4 = 2(4) + 3 = 11So the value of S4 = 5 + 7 + 9 + 11 = 32.
- Please listen carefully to Sir Roy for the answer and further discussion of the topic. Have a nice day everyone. :-)