Please solve Q 27 - 31. Q27. #a_n=na_{n-1}, a_1=1, a_3=?# Q28. Convergence region of #y = 1+x/2 + x^2/4 + ...# Q29. #sum_{r=1}^20( 3r+1)# Q30. Which is a field? #N, Z, Q, M_{2 times 2}# Q31, Inverse of #omega# in #{ 1, omega, omega^2 }#

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Answer 1 · 2018-06-28T17:41:54

#cancel{"That's pretty hard to read."}# I retyped in the question after I answered it but it was deleted.

Q27. #a_n=na_{n-1}, a_1=1#

#a_2 = 2 a_1 = 2#

#a_3 = 3 a_2 = 6#

#a_n = n!#

Q28. #y = 1+x/2 + x^2/4 + ...#

#y = sum_{n=0}^{infty} (x/2)^n #

That converges when #|x/2| < 1 or |x|< 2#

Q29. #sum_{r=1}^20( 3r+1)#

# = 3 sum_{r=1}^20 r+sum_{r=1}^20 1#

# = 3(20)(21)/2 + 20 =650#

Q30. #Q,# the rational numbers, is the primary and most important field in mathematics. Don't trust folks who tell you it's the reals.

Two by two matrices form a ring but there are non-zero matrices with zero determinants that are not invertible, so #M_{2 times 2}# isn't a field.

Q31. This seems like it's referring to #omega# the cube root of #-1.#

#omega^3 = 1#

#omega = 1/omega^2#

The multiplicative inverse of #omega# is #1/omega^2.#