How to find the common ratio, the common difference and the first four terms of a geometric sequence and a arithmetic sequences, if

Question

The first term of the geometric sequence is 4, while 2 is the first term of

an arithmetic sequence.

And, if the common ratio of the geometric sequence equals the common difference of the arithmetic sequence and their third terms are equal.

Answer 1

The formula for finding the geometric sequence is ar^n-1, likewise for an arithmetic formula, it is a + (n-1)d. We can find the ratio(r) and difference (d) by setting the two equal to each other and plugging in the numbers. the first terms (a) are given to us, so we will put 4 in the first term if the geometric sequence and put 2 for a in the arithmetic sequence formula. We also know that their third terms are equal. We will plug in 3 for n on both sides since that what determines which term we are calculating for.

4r^3-1 = 2 + (3-1)d

since we n on now r = d, replace them for the same variable and solve.

4x^3-1 = 2 + (3-1)x

Answer 2

GP: 4, 4r, 4r^2

AP 2, 2+d, 2+2d

r=d, 4r^2 = 2+2d

2r^2 = 1+d = 1+r

2r^2 -r -1 = 0

(2r+1)(r-1) = 0

r = -1/2 or 1

d = -1/2 or 1

GP: 4,-2, 1 or 4,4,4

AP: 2, 3/2, 1 or 2,3,4

4,-2,1,1/2 and 2,3/2,1,1/2 or 4,4,4,4 and 2,3,4,5