How to find the First Term of a sequence in the Sum of the Terms of Finite Geometric Sequence?
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How to find a Sum of the Terms if the Fist Term is missing?
Sn = a (r^n - 1)/(r-1)
so solve for a
The first term is never missing. If you have the sum of terms k+1 to n, then that is Sn - Sk and both of those include a.
To find the first term of a sequence in the sum of the terms of a finite geometric sequence, you can use the following formula:
S = a(1 - r^n) / (1 - r)
where:
- S is the sum of the terms
- a is the first term of the sequence
- r is the common ratio between the terms
- n is the number of terms in the sequence
If the first term is missing, but you know the sum of the terms and the common ratio, you can rearrange the formula to solve for the first term (a). Here's how:
1. Start with the formula for the sum of the terms:
S = a(1 - r^n) / (1 - r)
2. Rearrange the formula to solve for the first term (a):
a = S(1 - r) / (1 - r^n)
By plugging in the known values for S, r, and n, you can calculate the first term (a).
Keep in mind that for this formula to work correctly, the common ratio (r) should be between -1 and 1, and the number of terms (n) should be a positive integer.
To find the first term of a sequence in the sum of the terms of a finite geometric sequence, you need to have information about the common ratio (r), the number of terms (n), and the sum of the terms (S). The formula to find the first term (a) is:
a = S / (r^n - 1)
Here's how you can calculate it step by step:
1. Gather the information: You need to know the common ratio (r), the number of terms (n), and the sum of the terms (S).
2. Plug the values into the formula: Substitute the values of r, n, and S into the formula for the first term (a = S / (r^n - 1)).
3. Calculate the exponent: Raise the common ratio (r) to the power of the number of terms (n) and subtract 1.
4. Divide the sum by the result: Divide the sum of the terms (S) by the value obtained from step 3.
The resulting value is the first term of the geometric sequence.
Now, let's move on to finding the sum of the terms if the first term is missing. To do this, you need to have information about the common ratio (r), the number of terms (n), and the first term (a). The formula to find the sum (S) is:
S = a * (r^n - 1) / (r - 1)
Follow these steps to calculate the sum:
1. Gather the information: You need to know the common ratio (r), the number of terms (n), and the first term (a).
2. Plug the values into the formula: Substitute the values of a, r, and n into the formula for the sum (S = a * (r^n - 1) / (r - 1)).
3. Calculate the exponent: Raise the common ratio (r) to the power of the number of terms (n) and subtract 1.
4. Multiply the first term by the result: Multiply the first term (a) by the value obtained from step 3.
5. Divide by (r - 1): Divide the result from step 4 by the common ratio minus 1 (r - 1).
The resulting value is the sum of the terms in the geometric sequence.