2n2y2(2n3)(3y3)
-31. The rule is t(n) = -2n3 + 12n2 -19n + 14 where n = 1, 2, 3, ...
(n2 + 2n - 1) (n2 + 2n - 1) = n4 + 2n3 - n2 + 2n3 + 4n2 - 2n - n2 - 2n + 1 = n4 + 4n3 + 2n2 - 4n + 1 try with n = 5: (5 squared + 10 - 1) squared = 34 squared = 1156 with formula (5^4) + (4 *(5^3)) + (2 * (5^2)) - (4 * 5) + 1 = 625 + 500 + 50 - 20 + 1 = 1156
2n3
The simplest rule that will generate the 4 terms, and the nth term isUn = 2n2 + n + 3Then Sn = Sum for k = 1 to n of (2k2 + k + 3)= 2*sum(k2) + sum(k) + sum(3)= 2*n*(n+1)*(2n+1)/6 + n*(n+1)/2 + 3*n= (2n3 + 3n2 + n)/3 + (n2 + n)/2 + 3n= (4n3 + 9n2 + 23n)/6The simplest rule that will generate the 4 terms, and the nth term isUn = 2n2 + n + 3Then Sn = Sum for k = 1 to n of (2k2 + k + 3)= 2*sum(k2) + sum(k) + sum(3)= 2*n*(n+1)*(2n+1)/6 + n*(n+1)/2 + 3*n= (2n3 + 3n2 + n)/3 + (n2 + n)/2 + 3n= (4n3 + 9n2 + 23n)/6The simplest rule that will generate the 4 terms, and the nth term isUn = 2n2 + n + 3Then Sn = Sum for k = 1 to n of (2k2 + k + 3)= 2*sum(k2) + sum(k) + sum(3)= 2*n*(n+1)*(2n+1)/6 + n*(n+1)/2 + 3*n= (2n3 + 3n2 + n)/3 + (n2 + n)/2 + 3n= (4n3 + 9n2 + 23n)/6The simplest rule that will generate the 4 terms, and the nth term isUn = 2n2 + n + 3Then Sn = Sum for k = 1 to n of (2k2 + k + 3)= 2*sum(k2) + sum(k) + sum(3)= 2*n*(n+1)*(2n+1)/6 + n*(n+1)/2 + 3*n= (2n3 + 3n2 + n)/3 + (n2 + n)/2 + 3n= (4n3 + 9n2 + 23n)/6
5(25m3 + 2n3)
There are many possible answers. Some are A: Un = (2n3 - 15n2 + 34n - 18)/3 B: Un = 2 - mod(n, 2) C: Un = 1.5 + 0.5*(-1)n
2n2y2(2n3)(3y3)
The common name for a fatty acid with a c18 2n3 structure is alpha-linolenic acid. It is an essential omega-3 fatty acid that is important for proper brain function and reducing inflammation in the body.
The simplest polynomial, of an infinite number of possible solutions, is Un = (-2n3 + 12n2 - 7n + 6)/3
-31. The rule is t(n) = -2n3 + 12n2 -19n + 14 where n = 1, 2, 3, ...
37 Fit the cubic: Un = (2n3 - 9n2 + 16n + 6)/3 for n = 1, 2, 3, ...
There are many possible answers. One possibility is Un = (2n3 - 3n2 + n + 24)/6
There are infinitely many possible answers. The simplest cubic polynomial is Un = (2n3 - 11n2 + 27n - 14)/2 for n = 1, 2, 3, 4.
(n2 + 2n - 1) (n2 + 2n - 1) = n4 + 2n3 - n2 + 2n3 + 4n2 - 2n - n2 - 2n + 1 = n4 + 4n3 + 2n2 - 4n + 1 try with n = 5: (5 squared + 10 - 1) squared = 34 squared = 1156 with formula (5^4) + (4 *(5^3)) + (2 * (5^2)) - (4 * 5) + 1 = 625 + 500 + 50 - 20 + 1 = 1156
25* * * * *24Using t(n) = (2n3 - 9n2 + 25n - 6)/6 for n = 1, 2, 3, ...Take your pick. I have given the justification for my answer.
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