How many terms are in the arithmetic sequence 7, 0, −7, . . . , −175? Hint: a n = a 1 + d(n − 1), where a 1 is the first term and d is

Solve for n:
-175 = 7-7 (n-1)

-7 (n-1) = 7-7 n:
-175 = 7-7 n+7

Add like terms. 7+7  =  14:
-175 = 14-7 n

-175 = 14-7 n is equivalent to 14-7 n = -175:
14-7 n = -175

Subtract 14 from both sides:
(14-14)-7 n = -175-14

14-14 = 0:
-7 n = -175-14

-175-14 = -189:
-7 n = -189

Divide both sides of -7 n = -189 by -7:
(-7 n)/(-7) = (-189)/(-7)

(-7)/(-7) = 1:
n = (-189)/(-7)

The gcd of -189 and -7 is -7, so (-189)/(-7) = (-7×27)/(-7×1) = (-7)/(-7)×27 = 27:
Answer: |  n = 27

qHow many terms are in the arithmetic sequence \(7,\ 0,\ -7,\ \dots \ , \ -175 \)

\(\begin{array}{rcl} a_n &=& a_1 +(n-1)\cdot d\\ n-1 &=& \frac{ a_n-a_1 } {d}\\ n &=& 1+\frac{ a_n- a_1 }{d} \end{array} \)

\(a_n = -175 \qquad a_1 = 7 \qquad d=-7\\ n = 1 + \frac{-175-7}{-7}\\ n= 1+ 25+1\\ \mathbf{n=27}\)

The answer is A.