Question

If 7and 189 are the first and fourth terms of a geometric progression respectively; find the sum of the first three terms of the progression?

Answer 1

In a geometric progression (GP), the relationship between the terms can be described using the common ratio \( r \).

Let the first term \( a = 7 \) and the fourth term \( a_4 = 189 \). The fourth term can be expressed in terms of the first term and the common ratio as follows:

\[
a_4 = a \cdot r^3
\]

Substituting the known values:

\[
189 = 7 \cdot r^3
\]

To isolate \( r^3 \), divide both sides by 7:

\[
r^3 = \frac{189}{7} = 27
\]

Now, take the cube root of both sides to find \( r \):

\[
r = \sqrt[3]{27} = 3
\]

Now we have the common ratio \( r = 3 \). We can find the second and third terms of the GP:

1. The second term \( a_2 \):

\[
a_2 = a \cdot r = 7 \cdot 3 = 21
\]

2. The third term \( a_3 \):

\[
a_3 = a \cdot r^2 = 7 \cdot 3^2 = 7 \cdot 9 = 63
\]

Now we can calculate the sum of the first three terms:

\[
S = a + a_2 + a_3 = 7 + 21 + 63
\]
\[
S = 91
\]

Thus, the sum of the first three terms of the progression is \( \boxed{91} \).