Math  /  Algebra

QuestionThe third and fourth terms of an arithmetic sequence are the first and second terms of a geometric sequence. If the first two terms of the arithmetic sequence are 5,2 , then what is the fourth term of the geometric sequence?

Studdy Solution

STEP 1

1. The arithmetic sequence has the first two terms as 5 and 2.
2. The third and fourth terms of the arithmetic sequence are the first and second terms of a geometric sequence.
3. We need to find the fourth term of the geometric sequence.

STEP 2

1. Determine the common difference of the arithmetic sequence.
2. Find the third and fourth terms of the arithmetic sequence.
3. Use the third and fourth terms to find the common ratio of the geometric sequence.
4. Calculate the fourth term of the geometric sequence.

STEP 3

Determine the common difference of the arithmetic sequence.
The common difference d d of an arithmetic sequence can be found by subtracting the first term from the second term:
d=25=3 d = 2 - 5 = -3

STEP 4

Find the third and fourth terms of the arithmetic sequence.
The third term a3 a_3 is given by:
a3=a1+2d=5+2(3)=56=1 a_3 = a_1 + 2d = 5 + 2(-3) = 5 - 6 = -1
The fourth term a4 a_4 is given by:
a4=a1+3d=5+3(3)=59=4 a_4 = a_1 + 3d = 5 + 3(-3) = 5 - 9 = -4

STEP 5

Use the third and fourth terms to find the common ratio of the geometric sequence.
The first term of the geometric sequence is 1 -1 and the second term is 4 -4 . The common ratio r r is:
r=41=4 r = \frac{-4}{-1} = 4

STEP 6

Calculate the fourth term of the geometric sequence.
The third term of the geometric sequence g3 g_3 is:
g3=g1r2=(1)42=116=16 g_3 = g_1 \cdot r^2 = (-1) \cdot 4^2 = -1 \cdot 16 = -16
The fourth term of the geometric sequence g4 g_4 is:
g4=g3r=164=64 g_4 = g_3 \cdot r = -16 \cdot 4 = -64
The fourth term of the geometric sequence is:
64 \boxed{-64}

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