Answer:What is the value of n in the series, 1, 6, 16, 31, 51, n?
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We can calculate the difference between two neighboring numbers.
6−1=5,16−6=10,31−16=15,51−31=20
Then, we take the difference between the results.
10−5=5,15−10=5,20−15=5
Thus, the pattern is quadratic. The general quadratic form is an2+bn+c .
From the first three terms, we can infer that
a+b+c=1,4a+2b+c=6, and 9a+3b+c=16.
Then, we combine the first with the second equation and the second with the third equation. We will get 3a+b=5 and 5a+b=10 , which can be solved simultaneously.
5a−3a=5,2a=5 Thus, a=5/2 .
3(52)+b=5 Thus, b=−5/2 .
(52)+(−52)+c=1 Thus, c=1 .
Thus, the quadratic expression is (52)(n2)−(52)n+1 .
For the solution, we substitute 6 into the equation. Thus, the 6th term for the sequence is (52)(62)−(52)6+1=90−15+1=76.
Step-by-step explanation:
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