Question

The ninth term of a given geometric progression, with reason q , is 1792, and its fourth term is 56. Thus, calculate the fourth term of another geometric progression, whose ratio is q +1 and whose first term is equal to the first term of the first P.G. described.

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Jayne

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To find the fourth term of the second geometric progression, we need to find the first term.

Let the first term of the first geometric progression be 'a' and the common ratio be 'q'.

We are given that the fourth term of the first geometric progression is 56.

The nth term of a geometric progression is given by:

a_n=a\cdot(q)^{n-1}

So, using the formula for the nth term of a geometric progression, we have:

a \cdot q^3 = 56 \quad \Rightarrow (1)

We are also given that the ninth term of the first geometric progression is 1792.

So, using the same formula, we have:

a \cdot q^8 = 1792 \quad \Rightarrow (2)

Dividing equation (2) by equation (1), we get:

\frac{a \cdot q^8}{a \cdot q^3} = \frac{1792}{56}

Simplifying, we have:

q^5 = 32

Taking the 5th root of both sides, we get:

q = \sqrt[5]{32} = 2

Now that we know the value of 'q', we can go back to equation (1) and solve for 'a' to find the first term of the first geometric progression:

a \cdot q^3 = 56

a \cdot 2^3 = 56

a \cdot 8 = 56

a = \frac{56}{8} = 7

So, the first term of the given geometric progression is 7, and the ratio is 2.

Now, we need to find the fourth term of another geometric progression with ratio q +1 and the same first term (7) as the first geometric progression.

The fourth term of a geometric progression is given by:

a_4=a\cdot(q+1)^{4-1}

Substituting the values, we have:

a_4 = 7 \cdot (2 + 1)^{4-1}

a_4 = 7 \cdot 3^3

a_4 = 7 \cdot 27

a_4 = 189

Therefore, the fourth term of the second geometric progression is 189.

Answer: The fourth term of the second geometric progression is 189.

Let the first term of the first geometric progression be 'a' and the common ratio be 'q'.

We are given that the fourth term of the first geometric progression is 56.

The nth term of a geometric progression is given by:

So, using the formula for the nth term of a geometric progression, we have:

We are also given that the ninth term of the first geometric progression is 1792.

So, using the same formula, we have:

Dividing equation (2) by equation (1), we get:

Simplifying, we have:

Taking the 5th root of both sides, we get:

Now that we know the value of 'q', we can go back to equation (1) and solve for 'a' to find the first term of the first geometric progression:

So, the first term of the given geometric progression is 7, and the ratio is 2.

Now, we need to find the fourth term of another geometric progression with ratio q +1 and the same first term (7) as the first geometric progression.

The fourth term of a geometric progression is given by:

Substituting the values, we have:

Therefore, the fourth term of the second geometric progression is 189.

Answer: The fourth term of the second geometric progression is 189.

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