What is the lat term, in a series of 10 terms with a common ratio of 2, a first term (a1) if 1 and a summation of 1023
You have a,r,n
Tn = a*r^(n-1)
T10 = 1*2^9 = 512
To find the last term, we need to know the formula for the nth term (an) of a geometric sequence. The formula is given by:
an = a1 * r^(n-1)
where an is the nth term, a1 is the first term, r is the common ratio, and n is the number of terms.
In this case, we are given that a1 = 1 and the common ratio r = 2. We also know that the sum of the series (S) is 1023.
The formula for the sum of a geometric series (S) is given by:
S = a1 * (1 - r^n) / (1 - r)
We can rearrange this formula to find n (the number of terms):
1023 = 1 * (1 - 2^n) / (1 - 2)
Simplifying further:
1023 = (1 - 2^n) / (-1)
Multiplying both sides by -1:
-1023 = 1 - 2^n
Rearranging the equation:
2^n = 1024
Taking the logarithm of both sides with base 2:
n = log2(1024) = 10
So, the number of terms in the series is 10.
Now, we can substitute the values of a1, r, and n into the formula for the nth term:
an = a1 * r^(n-1)
= 1 * 2^(10-1)
= 1 * 2^9
= 1 * 512
= 512
Therefore, the last term of the series is 512.