Final answer:
To find the sequence based on the rule 'subtract 3' starting with 60, we successively subtract 3 from each previous number, resulting in the sequence: 60, 57, 54, 51, 48, 45, 42. Having two sequences {an} and {bm}, we add them by forming a new sequence out of sums of individual terms of these two sequences.
Step-by-step explanation:
The question asks us to create a sequence based on the rule 'subtract 3' starting with the number 60 and then write the next six numbers following this rule. To answer this problem, we will begin with the number 60 and consistently subtract three from each subsequent number to identify the pattern.
Start with 60.
Subtract 3 to get the next number: 60 - 3 = 57.
Continue the pattern: 57 - 3 = 54.
Next number: 54 - 3 = 51.
Following the pattern: 51 - 3 = 48.
Next in the sequence: 48 - 3 = 45.
And finally: 45 - 3 = 42.
Thus, the complete sequence based on the rule 'subtract 3' starting from 60 is: 60, 57, 54, 51, 48, 45, 42.
Having two sequences {an} and {bm}, we add them by forming a new sequence out of sums of individual terms of these two sequences. For this idea to work, the sequences have to be indexed in the same way. If one sequence's index starts earlier (for instance, if for {an} we have n = −1,0,1,2,3,... and for {bm} we have m = 2,3,4,...), we can always start the "late" sequence a bit earlier by adding zeroes at its beginning, in this example we would put b−1 = 0, b0 = 0, b1 = 0, and now we can add them. However, this problem does not happen too often.
In the following definition, we will for simplicity assume that the sequences are indexed in the same way and use the common index n for them.