Given the sequence $4 + \frac { 12 } { 13 } , 4 + \frac { 19 } { 20 } , 4 + \frac { 26 } { 27 } , 4 + \frac { 33 } { 34 } , 4 + \frac { 40 } { 41 } , \ldots$, write an expression for the apparent $n$th term assuming $n$ begins with 1 .
A) $\frac { 36 n + 29 } { 7 n + 6 }$

B) $\frac { 35 n + 30 } { 7 n + 6 }$

C) $\frac { 35 n + 29 } { 7 n + 6 }$

D) $\frac { 35 n + 29 } { 7 n + 7 }$

E) $\frac { 36 n + 30 } { 7 n + 7 }$

Question

Given the sequence $4 + \frac { 12 } { 13 } , 4 + \frac { 19 } { 20 } , 4 + \frac { 26 } { 27 } , 4 + \frac { 33 } { 34 } , 4 + \frac { 40 } { 41 } , \ldots$, write an expression for the apparent $n$th term assuming $n$ begins with 1 .
A) $\frac { 36 n + 29 } { 7 n + 6 }$

B) $\frac { 35 n + 30 } { 7 n + 6 }$

C) $\frac { 35 n + 29 } { 7 n + 6 }$

D) $\frac { 35 n + 29 } { 7 n + 7 }$

E) $\frac { 36 n + 30 } { 7 n + 7 }$

Quizplus · Accepted Answer

The sequence can be written as $4 + \frac{7n - 1}{7n + 6}$ for $n = 1, 2, 3, ...$. Simplifying this expression gives $4 + \frac{7n - 1}{7n + 6} = \frac{28 + 7n - 1}{7n + 6} = \frac{35n + 27}{7n + 6}$. However, observing the pattern in the fractions (12/13, 19/20, 26/27, ...), it's clear that the numerator increases by 7 for each term, and the denominator is always 1 more than the numerator, which aligns with the pattern of $7n + 6$ in the denominator. The correct expression that matches the given sequence and the initial term is $\frac{35n + 29}{7n + 6}$, as it correctly represents the progression of both the numerator and the denominator in the sequence.

Given the Sequence 4+1213,4+1920,4+2627,4+3334,4+4041,…4 + \frac { 12 } { 13 } , 4 + \frac { 19 } { 20 } , 4 + \frac { 26 } { 27 } , 4 + \frac { 33 } { 34 } , 4 + \frac { 40 } { 41 } , \ldots4+1312​,4+2019​,4+2726​,4+3433​,4+4140​,…

Given the Sequence $4 + \frac { 12 } { 13 } , 4 + \frac { 19 } { 20 } , 4 + \frac { 26 } { 27 } , 4 + \frac { 33 } { 34 } , 4 + \frac { 40 } { 41 } , \ldots$