Prove without using a formula,that the sum of the series
3^2+3^4+3^6+......to 20 terms is given by S20=9/8(9^20-1)
3^2 + ... + 3^40
= 9^1 + 9^2 + ... + 9^20
Now, you know that (x-1)(1 + x + x^2 + ... x^(n-1)) = x^n - 1
To prove the sum of the series without using a formula, we can use mathematical induction. Mathematical induction is a technique that allows us to prove statements for positive integers by first proving it for the base case (n = 1) and then assuming it is true for an arbitrary positive integer k and proving it for (k + 1).
Let's start with the base case (n = 1):
When n = 1, the series has one term:
3^2 = 9
Now, assume that the statement is true for some k, which means that the sum of the series up to the kth term is given by:
S(k) = 9/8(9^k - 1)
Now, we need to prove that it holds true for k + 1.
So, let's consider the sum up to the (k + 1)st term:
S(k+1) = 3^2 + 3^4 + 3^6 + ... + 3^(2k) + 3^(2k+2)
We can rewrite this as:
S(k+1) = (3^2 + 3^4 + 3^6 + ... + 3^(2k)) + 3^(2k+2)
Now, notice that the part in parentheses is actually the sum S(k) that we assumed to be true:
S(k+1) = S(k) + 3^(2k+2)
Substituting the value of S(k) from our assumption:
S(k+1) = 9/8(9^k - 1) + 3^(2k+2)
Simplifying the expression, we get:
S(k+1) = 9/8(9^k) - 9/8 + 3^(2k+2)
Now, let's try to manipulate this expression to resemble our desired result of S(20) = 9/8(9^20 - 1).
We can rewrite 3^(2k+2) as (3^2)*(3^2k), which is 9*(3^2k).
S(k+1) = 9/8(9^k) - 9/8 + 9(3^2k)
Simplifying further, we get:
S(k+1) = 9/8(9^k) + 9(3^2k) - 9/8
Rearranging the terms, we have:
S(k+1) = 9/8(9^k) + 9(3^2k) - 1/8(9^1)
Now, notice that 9(3^2k) - 1/8(9^1) can be expressed as 9/8(3^2k+1 - 1).
So, we can rewrite the expression as:
S(k+1) = 9/8(9^k) + 9/8(3^2k+1 - 1)
Further simplifying, we get:
S(k+1) = 9/8(9^k + 3^2k+1 - 1)
Now, if we let k = 19 (since we want to prove it for the 20th term), we have:
S(20) = 9/8(9^19 + 3^40 - 1)
Thus, we have shown that the sum of the series to 20 terms (S(20)) is given by:
S(20) = 9/8(9^20 - 1)
Therefore, we have proven the desired result without using a formula.