2022 AMC 10 A

Complete problem set with solutions and individual problem pages

Problem 20 Easy

A four-term sequence is formed by adding each term of a four-term arithmetic sequence of positive integers to the corresponding term of a four-term geometric sequence of positive integers.

The first three terms of the resulting four-term sequence are 57, 60, and 91. What is the fourth term of this sequence?

  • A.

    190

  • B.

    194

  • C.

    198

  • D.

    202

  • E.

    206

Answer:E

Arithematic sequence: a, a+d,a+2d,a+3d.

Geometric sequence: x,xr,xr^2,xr^3.

\begin{cases}a+x=57 \\ a+d+xr=60 \\ a+2d+xr^2=91 \end{cases}

Then, \begin{cases} d+(r-1)x=3\\ d+(r^2-r)x=31 \end{cases}

So, (r^2-2r+1)x=28

(r-1)^2x=28

(r-1)\sqrt{x}=2\sqrt{7}

Since x is an integer, x has factor of 7.

Let x=7m^2 (m is an interger), q=\frac2m+1.

\because a+x=57, ax>0

\therefore m=1 or 2.

When m=1, q=2+1=3.

Arithematic sequence: 7,21,63,189.

Geometric sequence: 50,39,28,17.

189+17=206.

When m=2, q=1+1=2.

Arithematic sequence: 28,56,112.

Geometric sequence: 22,4,-16 (contradiction).

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