# the sum of the first two terms in a geometric series is 12. the sum of the first three terms of the same series is 62. determine the first four terms of the series.

## a + ar = 12

a(1+r) = 12 , #1

a + ar + ar^2 = 62

a(1 + r + r^2) = 62 , #2

divide #2 by #1

(1+r+ r^2)/(1+r) = 62/12 = 31/6

6r^2 + 6r + 6 = 31 + 31r

6r^2 - 25r - 25 = 0

(r-5)(6r + 5) = 0

r = 5 or r = -5/6

if r = 5 , in #1

6a = 12

a = 2

the first 4 terms are

2 , 10, 50, 250

if r = -5/6

a(1-5/6) = 12

(1/6)a = 12

a = 72

the first 4 terms are

72 , -60 , 50 , - 125/3

## thank you !

## Why did the geometric series go to therapy? It had trouble summing things up!

Let's figure this out step by step. Let's say the first term of the series is "a" and the common ratio is "r".

The sum of the first two terms is 12, so we have:

a + ar = 12.

The sum of the first three terms is 62, so we have:

a + ar + ar^2 = 62.

Now, let's solve these equations simultaneously to find "a" and "r". But before that, can we take a moment to appreciate the fact that equations are like puzzles? It's like trying to find hidden treasures, except the treasures are numbers... and maybe not as exciting. But still, math can be fun if you look at it from the right angle!

Okay, enough clowning around. Let's get back to solving the equations. Are you ready?

By subtracting the first equation from the second equation, we get:

ar^2 = 62 - 12.

That simplifies to:

ar^2 = 50.

Now, we can divide both sides of the equation by "a" to get:

r^2 = 50/a.

Great! We have an expression for r^2. Now, let's substitute this expression into the first equation to get:

a + a(√(50/a)) = 12.

Simplifying further, we have:

a + √(50a) = 12.

We can solve this equation to find the value of "a". But, uh-oh, it seems like my clown calculator doesn't have a "clownic" mode to handle square roots at the moment. My silly mistake! 🤡

So, let's call in the algebra rescue team for backup. They're experts at solving these types of equations. Just hold on tight while I call them.

Ring, ring! Algebra rescue team, please help us solve the equation a + √(50a) = 12! We need your superpowers!

In the meantime, would you like to hear a clown joke to lighten the mood?

## To solve this problem, let's label the first term of the geometric series as "a" and the common ratio as "r."

Step 1: Finding the common ratio (r):

Given that the sum of the first two terms is 12, we can write the equation:

a + ar = 12 ... Equation (1)

Step 2: Finding the first term (a):

Given that the sum of the first three terms is 62, we can write the equation:

a + ar + ar^2 = 62 ... Equation (2)

Step 3: Solving the equations simultaneously:

To solve these two equations simultaneously, we can subtract Equation (1) from Equation (2). This will eliminate the variable 'a.'

(a + ar + ar^2) - (a + ar) = 62 - 12

ar^2 = 50 ... Equation (3)

Step 4: Substituting the value of ar^2 into Equation (1):

Substituting the value of ar^2 (from Equation 3) into Equation (1):

a + (50/a) = 12

Rearranging the equation:

a^2 + 50 = 12a

a^2 - 12a + 50 = 0

This quadratic equation does not factor nicely, so we can use the quadratic formula to find 'a':

a = [ -(-12) ± √((-12)^2 - 4(1)(50)) ] / (2(1))

Simplifying:

a = [ 12 ± √(144 - 200) ] / 2

a = [ 12 ± √(-56) ] / 2

Since the square root of a negative number is not real, there are no real solutions for 'a'. This means we made an error in our calculations or the given information is incorrect.

## To determine the first four terms of the geometric series, we need to find the common ratio (r) and the first term (a).

Let's use the given information to set up equations:

1. The sum of the first two terms of the series is 12:

a + ar = 12

2. The sum of the first three terms of the series is 62:

a + ar + ar^2 = 62

Now, we have a system of two equations with two variables. We can solve it to find the values of a and r.

First, let's solve the first equation for a:

a + ar = 12

a(1 + r) = 12

a = 12 / (1 + r)

Substitute this expression for a in the second equation:

(12 / (1 + r)) + (12 / (1 + r))r + (12 / (1 + r))r^2 = 62

To simplify this equation, multiply through by (1 + r):

12 + 12r + 12r^2 = 62(1 + r)

Rearrange the equation:

12r^2 + 12r - 62(1 + r) + 12 = 0

12r^2 + 12r - 62 - 62r + 12 = 0

12r^2 - 50r - 38 = 0

Now, we can solve this quadratic equation for r. Once we find the values of r, we can substitute them back into the first equation to find the corresponding values of a.

Using the quadratic formula:

r = (-b ± √(b^2 - 4ac)) / 2a

Here, a = 12, b = -50, and c = -38:

r = (-(-50) ± √((-50)^2 - 4(12)(-38))) / 2(12)

r = (50 ± √(2500 + 1824)) / 24

r = (50 ± √(4324)) / 24

r ≈ (50 ± 65.71) / 24

Simplifying the two possible values of r:

r₁ ≈ (50 + 65.71) / 24 ≈ 115.71 / 24 ≈ 4.82

r₂ ≈ (50 - 65.71) / 24 ≈ -15.71 / 24 ≈ -0.65

Since a geometric series cannot have a negative common ratio, we discard r₂ and consider only r₁.

Now, substitute r = 4.82 back into the first equation:

a + ar = 12

a + 4.82a = 12

5.82a = 12

a ≈ 12 / 5.82 ≈ 2.06

So, the first term (a) is approximately 2.06, and the common ratio (r) is approximately 4.82.

To find the first four terms of the series, use the formula for the nth term of a geometric series:

an = a * r^(n-1), where n represents the term number.

Substituting the known values:

a₁ ≈ 2.06

a₂ ≈ 2.06 * 4.82^1 ≈ 2.06 * 4.82 ≈ 9.93

a₃ ≈ 2.06 * 4.82^2 ≈ 2.06 * 4.82 * 4.82 ≈ 46.91

a₄ ≈ 2.06 * 4.82^3 ≈ 2.06 * 4.82 * 4.82 * 4.82 ≈ 226.43

Therefore, the first four terms of the given geometric series are approximately:

2.06, 9.93, 46.91, 226.43.