One sequence of alternating terms of
the series 1+2+3+4+5+8+... forms an
arithmetic progression, while the
other sequence of alternating terms
forms a geometric progression. Sum
the first 10 terms of each progression
and hence find the sum of the first 20
terms of the series.
clearly,
1,3,5 are an AP
2,4,8 are a GP
Now just plug in your formulas for partial sums.
I've found the first 10 terms of each progression, but m having problems with sum of the first 20 terms of the series. Am I going to solve for A.P and G.P differently.
To solve this problem, we need to identify the two alternating sequences within the given series.
Let's start by listing the terms:
1 + 2 + 3 + 4 + 5 + 8 + ...
We can see that the sequence starts with the natural numbers and then skips to the powers of 2. Hence, the series can be split into two alternating sequences as follows:
Arithmetic progression: 1, 3, 5, ...
Geometric progression: 2, 4, 8, ...
Now, let's find the sum of the first 10 terms of each progression.
1. Arithmetic progression:
To find the sum of an arithmetic progression, we can use the formula:
Sum = (First term + Last term) * Number of terms / 2
Here, the first term is 1, and the common difference is 2. So the last term can be found using the formula:
Last term = First term + (Number of terms - 1) * Common difference
Last term = 1 + (10 - 1) * 2 = 19
Now, we can calculate the sum of the arithmetic progression:
Sum (AP) = (First term + Last term) * Number of terms / 2
= (1 + 19) * 10 / 2
= 20 * 10 / 2
= 200 / 2
= 100
Therefore, the sum of the first 10 terms of the arithmetic progression is 100.
2. Geometric progression:
To find the sum of a geometric progression, we can use the formula:
Sum = (First term * (1 - Common ratio^n)) / (1 - Common ratio)
Here, the first term is 2, and the common ratio is 2 (as each term is double the previous). The exponent (n) represents the number of terms. So we can calculate the sum of the geometric progression as:
Sum (GP) = (First term * (1 - Common ratio^n)) / (1 - Common ratio)
= (2 * (1 - 2^10)) / (1 - 2)
= (2 * (1 - 1024)) / (1 - 2)
= (2 * (-1023)) / -1
= -2046
Therefore, the sum of the first 10 terms of the geometric progression is -2046.
Now, let's find the sum of the first 20 terms of the series:
Since we have identified that the given series is formed by alternating terms from an arithmetic progression and a geometric progression, we can find the sum of the first 20 terms by summing the individual sums.
Sum (series) = Sum (AP) + Sum (GP)
= 100 + (-2046)
= -1946
Therefore, the sum of the first 20 terms of the given series is -1946.
To find the sum of the first 10 terms of each progression, we need to understand the patterns in both the arithmetic and geometric progressions.
In the given series, the alternating terms form an arithmetic progression. This means that the difference between any two consecutive terms is constant. Let's denote this common difference as 'd'. From observation, we can see that the first term of the arithmetic progression is 1, the second term is 3 (1+2), the third term is 5 (3+2), the fourth term is 7 (5+2), and so on. The common difference can be calculated as d = (second term - first term) = 3 - 1 = 2.
Now, let's calculate the sum of the first 10 terms of the arithmetic progression using the formula for the sum of an arithmetic progression:
Sum_arithmetic = (n/2) * (2a + (n-1)d)
where n is the number of terms, a is the first term, and d is the common difference.
Using the given information, we substitute n = 10, a = 1, and d = 2 into the formula:
Sum_arithmetic = (10/2) * (2*1 + (10-1)*2)
= 5 * (2 + 9*2)
= 5 * (2 + 18)
= 5 * 20
= 100
So, the sum of the first 10 terms of the arithmetic progression is 100.
Now, let's focus on the other alternating terms that form a geometric progression. In a geometric progression, each term is obtained by multiplying the previous term by a constant ratio. Let's denote this common ratio as 'r'. From observation, we can notice that the first term of the geometric progression is 2 (2 x 1), the second term is 8 (2 x 4), the third term is 32 (8 x 4), the fourth term is 128 (32 x 4), and so on. The common ratio can be calculated as r = (second term / first term) = 8 / 2 = 4.
To find the sum of the first 10 terms of the geometric progression, we will use the formula for the sum of a geometric progression:
Sum_geometric = a * (1 - r^n) / (1 - r)
where a is the first term, r is the common ratio, and n is the number of terms.
Using the given information, we substitute a = 2, r = 4, and n = 10 into the formula:
Sum_geometric = 2 * (1 - 4^10) / (1 - 4)
= 2 * (1 - 1048576) / (1 - 4)
= 2 * (-1048575) / (-3)
= 2 * 349525 / 3
= 232350 / 3
= 77450
So, the sum of the first 10 terms of the geometric progression is 77450.
Finally, to find the sum of the first 20 terms of the series, we add the sum of the first 10 terms of the arithmetic progression and the sum of the first 10 terms of the geometric progression:
Sum_series = Sum_arithmetic + Sum_geometric
= 100 + 77450
= 77550
Therefore, the sum of the first 20 terms of the series is 77550.