if p is the nth term
8+20+52+.....+p
evaluate the sum to n-of the series
plz show step
Looking at only 3 terms,
they are not in AP
they are not in GP
looking for some pattern:
8 = 4*2
20 = 4*5
52 = 4*13
I see nothing special about 2, 5, 13
2 = 1+1
5 = 4+1
13 = 12+1
still nothing
There are lots of sequences that fit those three numbers. Possibly the simplest is a quadratic,
10x^2-18x+16
So,
p
∑ 10x^2-18x+16 = 2/3 p(5p^2-6p+13)
k=1
Of course, why didn't I think of that
2 data values --- linear
3 data values --- quadratic
4 data values --- cubic
etc
To evaluate the sum of the given series, we need to find a pattern or equation for the nth term (p) and then use it to calculate the sum.
Let's analyze the given series: 8+20+52+...
If we observe closely, we can see that each term is obtained by multiplying the previous term by 2 and then adding a constant number. Let's break it down:
To go from 8 to 20, we multiply 8 by 2 and add 4: (8 * 2) + 4 = 20
To go from 20 to 52, we multiply 20 by 2 and add 12: (20 * 2) + 12 = 52
So, it seems like each term is obtained by multiplying the previous term by 2 and adding a constant number.
To find this constant number, let's subtract each term from its corresponding term:
20 - 8 = 12
52 - 20 = 32
We can see that the difference between these two terms is 12, indicating that our constant number is 12.
Now, we have the relationship between the terms: p = (previous term * 2) + 12.
To find the sum of the series, we can use the formula for the sum of an arithmetic series:
S = (n/2) * (a + l)
Where:
- S is the sum of the series,
- n is the number of terms in the series,
- a is the first term,
- l is the last term.
In our case, the first term is 8, and the nth term is p. We need to find the last term.
To find the last term, we can substitute p into the equation: p = (previous term * 2) + 12.
So, the nth term is (p * 2) + 12.
Since the last term is p, we can equate it with (p * 2) + 12:
p = (p * 2) + 12
Now, we can solve this equation to find the value of p.
p = (p * 2) + 12
p - 2p = 12
-p = 12
p = -12
Therefore, the last term (l) is -12.
Now, we can substitute the given values into the formula for the sum of an arithmetic series:
S = (n/2) * (a + l)
= (n/2) * (8 + (-12))
Simplifying:
S = (n/2) * (-4)
= -2n
So, the sum of the series is -2n.
Note: It's important to double-check the calculations and ensure the provided solution is accurate.