The 2nd And 4th Term Of A G.P Are 8 And 32 Respectively What Is The Sum Of The First And Four Term

4, 8, 16, 32

Do you see the pattern?

To find the sum of the first and fourth term of a geometric progression (G.P), we need to determine the common ratio (r) first.

Given that the second term (a2) is 8 and the fourth term (a4) is 32, we can use the formula for the nth term of a G.P to find the common ratio:

an = a1 * r^(n-1)

For a2:
8 = a1 * r^(2-1)
8 = a1 * r
a1 = 8/r

For a4:
32 = a1 * r^(4-1)
32 = a1 * r^3

Now we can substitute the value of a1 in terms of r into the equation for a4:

32 = (8/r) * r^3
32 = 8 * r^2

Divide both sides by 8:

4 = r^2

Take the square root of both sides:

r = ±2

We have two possible values for r: 2 and -2. Let's consider both cases:

1. r = 2:
Using the formula for the sum of the first n terms of a G.P, we can calculate the sum of the first and fourth term:

S = a1 * (1 - r^n) / (1 - r)

Substituting the values we know:

S = (8/2) * (1 - 2^4) / (1 - 2)
S = 4 * (1 - 16) / -1
S = 4 * (-15) / -1
S = -60

Therefore, when r = 2, the sum of the first and fourth term is -60.

2. r = -2:
Using the same formula as before, we can calculate the sum of the first and fourth term:

S = a1 * (1 - r^n) / (1 - r)

Substituting the values we know:

S = (8/-2) * (1 - (-2)^4) / (1 - (-2))
S = -4 * (1 - 16) / (1 + 2)
S = -4 * (-15) / 3
S = 60 / 3
S = 20

Therefore, when r = -2, the sum of the first and fourth term is 20.

In conclusion, the sum of the first and fourth term of the geometric progression can be either -60 or 20, depending on the value of the common ratio (r).

To find the sum of the first and fourth terms of the geometric progression (G.P.), you need to determine the common ratio (r) first.

In a G.P., the general form of the nth term is given by the formula:
an = a1 * r^(n-1).

Given that the second term (a2) is 8 and the fourth term (a4) is 32, we can set up two equations using the formula above:
a2 = a1 * r^(2-1) --> 8 = a1 * r
a4 = a1 * r^(4-1) --> 32 = a1 * r^3.

Divide the second equation by the first equation:
32 / 8 = (a1 * r^3) / (a1 * r) --> 4 = r^2.

By taking the square root of both sides of the equation, we find:
r = ±2.

Since a G.P. cannot have a negative common ratio, we take r = 2.

Now, we can calculate the first term (a1) by substituting the value of r into the first equation:
8 = a1 * 2 --> a1 = 4.

So, the first term (a1) is 4, and the fourth term (a4) is 32. To find the sum of the first and fourth terms, we add them together:

4 + 32 = 36.

Therefore, the sum of the first and fourth terms of the geometric progression is 36.