Please solve Q 27 - 31. Q27. $a_{n} = n a_{n - 1}, a_{1} = 1, a_{3} = ?$ $a_n=na_{n-1}, a_1=1, a_3=?$ 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 $1/omega^2.$