For some arithmetic progression, it is known that its tenth term is 191, and the twentieth is 371.

For some arithmetic progression, it is known that its tenth term is 191, and the twentieth is 371. Calculate the 5th term of a geometric progression if its first term is equal to the first term of this arithmetic progression, and the denominator is 4.

Let’s calculate the 1st term of the arithmetic progression using the formulas.

A1 = A10 – 9d.

A1 = A20 – 19d.

Let’s equate the right-hand sides of the two equations, substitute the values A10 = 191 and A20 = 371, simplify and find d – the difference of the arithmetic progression.

A10 – 9d = A20 – 19d.

191 – 9d = 371 – 19d.

10d = 180.

d = 18.

Let’s calculate the 1st term.

A1 = 191 – 9 * 18.

A1 = 29.

Now let’s calculate the 5th term B5 of the geometric progression if B1 = A1 = 29 and the denominator q = 4.

B5 = B1 * q ^ 4.

B5 = 29 * 4 ^ 4.

B5 = 7424.

Answer: The 5th term of the geometric progression is 7424.