Question

What will be the first term of a geometric series which has its sum 280, common ratio 3 and the last term 189 ?

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Answer 1

The first term of the geometric series is approximately 10.37.

Step-by-step explanation:

To find the first term of a geometric series, we can use the formula:

a = S / (r^n)

Where a is the first term, S is the sum of the series, r is the common ratio, and n is the number of terms.

In this case, the sum (S) is 280, the common ratio (r) is 3, and the last term is 189.

Let's substitute these values into the formula:

a = 280 / (3^n) = 189

By simplifying the equation, we get:

3^n = 280 / 189

To solve for n, we can take the logarithm of both sides:

n = log(280 / 189) / log(3)

Using a calculator, we find that n is approximately 2.0054.

Since n represents the number of terms and must be a whole number, we can round up to 3.

Now, let's substitute the value of n back into the equation to find the first term:

a = 280 / (3^3) = 280 / 27 = 10.37

Therefore, the first term of the geometric series is approximately 10.37.

Answer 2

b1=7

Step-by-step explanation:

If we deal with geometric series, we should use the formulas for it. If there is the last term, it is not infinite geometric series. The sum of geometric series which isn't infinite is equal to b1(r^n-1)/(r-1) where b1 is the first term and r is the common ratio.

So 280= b1(3^n-1)/(3-1)

560= b1(3^n-1)

560= b1*3^n-b1

Then express bn=b1*r^(n-1)

189= b1*3^(n-1)

189= b1*3^n*1/3

567= b1*3^n when

560= b1*3^n-b1

560=567-b1

b1=7