For sequences with the same positive first term

MATHEMATICS · Single correct · AP · AP Precalculus · Exponential and Logarithmic Functions · Exponential and Logarithmic Functions

Problem

a

, which statement correctly compares the

n

th terms of an arithmetic progression (common difference

d>0

) and a geometric progression (common ratio

r>1

) as

n

increases? (a) The

n

th term of the AP eventually exceeds that of the GP because the constant difference

d

accumulates faster than a small ratio

r

. (b) The

n

th term of the GP grows exponentially faster than that of the AP. (c) Both the AP and GP grow at the same overall rate when

r

is close to

1

, so neither ultimately dominates. (d) The GP overtakes the AP only if

r>\dfrac{a}{d}

; otherwise the AP stays larger for all

n