Respuesta :
Answer: The 75th term is 86
Step-by-step explanation:
Since the sequence consists of all natural numbers which are neither perfect squares nor perfect cubes, we can write out the next ninety numbers starting from 2, and then ruling out the squares and cubes. We then count to the 75th number to the desired term. This is shown in the attachment below.
We can as well sum the number of squares and cubes in the range with the number of the number of the desired term i.e
Number of squares = 8 ( 4,9,16,25,36,49,64,81)
Number of cubes = 2 (8 and 27; excluding 64 which is as well a square)
Desired term = 75
So, 8+2+75 = 85.
Therefore, the 85th term counting from 2 ( i.e 86) is the 75th term of the sequence.
The 75th of term of the sequence excluding perfect squares and cubes is 86.
Sequence = 2, 3, 5, 6, 7, 10, ...
Given that the sequence is a series of natural numbers starting from 2. Therefore, the 75th term without excluding perfect squares and cubes will be 76 = (75+1) as 1 is not included in series.
As the 75th term is number 76. Thus,
Perfect squares till 76 are,
4, 9, 16, 25, 36, 49, 64
Perfect cubes till 76 are,
8, 27
Note: As 64 is already included, we not include 64 this time.
Thus, A total of 9 numbers are added to 76, but as 81 is (9²). A total of 10 numbers will be added to 76. therefore, the final number is 86 = 76+10.
Hence, the 75th of term of the sequence excluding perfect squares and cubes is 86.
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