If A1 and A2 are known to be equal difference sequence and A1 = 10, their arithmetic mean is 53.5, then A13 is equal to I figured out 46, but every option was 46 a1 a2 a3 …… A30 is the arithmetic mean of arithmetic difference sequence

If A1 and A2 are known to be equal difference sequence and A1 = 10, their arithmetic mean is 53.5, then A13 is equal to I figured out 46, but every option was 46 a1 a2 a3 …… A30 is the arithmetic mean of arithmetic difference sequence

a(n)=10+(n-1)d,
a(2)=10+d,
53.5 = [a(1)+a(2)]/2 = [20+d]/2,d= 107 - 20 = 87.
a(n)=10+87(n-1).
a(13)=10+87*12=1054.

In the expansion of (x ^ 3 + 1 / x ^ 2) ^ n, only the sixth term has the largest coefficient, then n =?
With binomial theorem solution, 3Q!

1
eleven
one hundred and twenty-one
one thousand three hundred and thirty-one
1,5,10,10,5,1
1,6,15,20,15,6,1
From Yang Hui's triangle, we can see that the coefficient of (a + b) ^ n expansion is the largest ([n / 2] + 1). (when n is an odd number, there is also a ([n / 2] + 2) term which is also large. [x] is the integral part of X.)
When [n / 2] + 1 = 6, n = 10 or 11
If the maximum coefficient has only one term, then n = 10

If (x ^ 2 - (1 / x)) ^ n expansion contains the sixth term, find the coefficient containing the fourth term

T6=C(n,5)*(x²)^(n-5)*(-1/x)^5
The exponent of X is 2 (N-5) - 5 = 2n-15 = 1
∴ n=8
The general term of (x ^ 2 - (1 / x)) ^ n expansion is t (R + 1)
∴ T(r+1)=C(8,r)*(x²)^(8-r)*(-1/x)^r
The exponent of X is 2 (8-r) - r = 16-3r = 4
∴ r=4
The coefficient of T5 is C (8,4) * - 1) ^ 4 = C (8,4) = 8 * 7 * 6 * 5 / (1 * 2 * 3 * 4) = 70
The coefficient of x ^ 4 is 70