given the geometric series 1+2+4+8 find the sum between (inclusively) the 5th term and the 15th term.
what I did I'm sure its wrong but I tired like that :
the 5th term is 16 because the r is *2 in each term and the a5th term is :
15=1*2^n
15=2^7.5
S15=1{(2)^7.5)-1}/2-1
S15=180
is that correct, I'm sure is not ,help please!!!
To find the sum of terms between the 5th and 15th term of the geometric series 1+2+4+8, you need to first determine the formula for the nth term of the series.
In a geometric series, where the common ratio between consecutive terms is constant, the formula for the nth term is:
an = a1 * r^(n-1)
Here, a1 is the first term of the series, r is the common ratio, and n is the term number.
In this case, a1 = 1 and r = 2.
Now, let's find the value of the 5th term (a5):
a5 = a1 * r^(5-1)
= 1 * 2^4
= 1 * 16
= 16
Similarly, let's find the value of the 15th term (a15):
a15 = a1 * r^(15-1)
= 1 * 2^14
= 1 * 16384
= 16384
Next, let's find the sum of terms between the 5th and 15th term (inclusive). We can use the formula for the sum of terms in a geometric series:
Sn = a1 * (1 - r^n) / (1 - r)
where Sn is the sum of the first n terms.
For this problem, we want to find the sum of terms from the 5th (a5) to the 15th (a15), so n = 15 - 5 + 1 = 11.
S11 = a5 * (1 - r^11) / (1 - r)
= 16 * (1 - 2^11) / (1 - 2)
= 16 * (1 - 2048) / (1 - 2)
= 16 * (-2047) / (-1)
= 32752
Therefore, the sum of the terms between the 5th and 15th term (inclusive) of the geometric series 1+2+4+8 is 32752.
To find the sum of a geometric series, you need to use the formula:
S_n = a * (r^n - 1) / (r - 1)
where:
S_n is the sum of the series up to the nth term,
a is the first term of the series,
r is the common ratio, and
n is the number of terms.
In your case, the series is 1+2+4+8.
We need to find the sum from the 5th term to the 15th term, which means we need to find the sum S_15 and subtract the sum S_4.
To find S_4, we can use the formula mentioned earlier:
S_4 = a * (r^4 - 1) / (r - 1)
Substituting the given values, we get:
S_4 = 1 * (2^4 - 1) / (2 - 1)
S_4 = 15
Now, let's find S_15:
S_15 = a * (r^15 - 1) / (r - 1)
Substituting the given values, we get:
S_15 = 1 * (2^15 - 1) / (2 - 1)
S_15 = 32767
Finally, to find the sum between the 5th term and the 15th term, we subtract S_4 from S_15:
Sum = S_15 - S_4
Sum = 32767 - 15
Sum = 32752
So, the sum between the 5th term and the 15th term of the geometric series 1+2+4+8 is 32,752.
Since
a = 1
r = 2
S4 = 1(2^4-1)/(2-1) = 15
S15 = 2^15-1 = 32768
So, the sum from S5 to S15 = S15-S4 = 32753