Math fast , i do not understand calculate a53 for the sequence (an)=(-7,-11/2,-4,-5/2,-1) a.)145/2 b.)71 c.)154/2 d.)74 calculate s68 for the arithmetic sequence (an)=(2/3n+1/6) a.)87/2 b.)273/6 c.)139/3 d.)4726/3

Given the sequence An = -7,-11/2,-4,-5/2,-1

It can be seen that the sequence increase by 3/2 (i.e. second term - first term = -11/2 - (-7) = -11/2 + 7 = 3/2

Thus, the common difference of the sequence is 3/2.

The formular for the nth term of a sequence is given by An = a + (n - 1)d where a is the first term of the sequence and d is the common difference.

Thus, the 53rd term of the sequence is given by

A(53) = -7 + (53 - 1)(3/2) = -7 + 52(3/2) = -7 + 78 = 71.

Therefore, A(53) = 71.

Part B:

The sum of the  nth term of a sequence is given by

Sn = n/2(2a + (n - 1)d) or Sn = n/2(a + l) where a is the first term, d is the common difference and l is the last term (or the nth term).

Given the sequence:

An = 2/3n + 1/6

The first term is given by An = 2/3(1) + 1/6 = 2/3 + 1/6 = 5/6
The last term (the 68th term) is given by An = 2/3(68) + 1/6 = 136/3 + 1/6 = 91/2

Thus, the sum of the first 68 terms (S68) is given by

S68 = 68/2(5/6 + 91/2) = 34(139/3) = 4726/3