What is difference between the sum of tenth terms of a the terms of the AP and GP whose first terms are -2 and 4
for the AP:
sum of 10 terms = 5(-4 + 9d) = -20 + 45d
for the GP:
sum of 10 terms = 4(r^10 - 1)/(r-1)
so I guess their difference would be 4(r^10 - 1)/(r-1) - (45d - 20)
strange question! Am I missing something here?
10
Oh, the difference between those is definitely worth a laugh!
Let's start with the Arithmetic Progression (AP). We'll find the sum of its 10 terms using the formula Sn = (n/2)(2a + (n-1)d), where a is the first term and d is the common difference. Plugging in the values, we get:
SnAP = (10/2)(2(-2) + (10-1)d)
= 5(-4 + 9d)
= -20 + 45d
Now, let's calculate the sum of the Geometric Progression (GP). The formula for that is S = a(1 - r^n)/(1 - r), where a is the first term and r is the common ratio. Plugging in the values, we get:
SnGP = 4(1 - r^10)/(1 - r)
So, the difference between the sums will be:
Difference = SnAP - SnGP
= (-20 + 45d) - 4(1 - r^10)/(1 - r)
I'm afraid I can't simplify it further, but hey, at least we had some fun getting to this point, right?
To find the difference between the sum of the tenth terms of an arithmetic progression (AP) and a geometric progression (GP) with given first terms, we need to first find the formulas for the AP and GP, and then calculate the sums.
AP Formula:
The formula for the nth term of an arithmetic progression is given by:
an = a1 + (n - 1) * d
where:
an = nth term of the AP
a1 = first term of the AP
d = common difference of the AP
In this case, the first term of the AP is -2:
a1 = -2
Since the question doesn't specify the common difference "d," we cannot directly calculate the sum of the tenth term of the AP.
GP Formula:
The formula for the nth term of a geometric progression is given by:
an = a1 * r^(n - 1)
where:
an = nth term of the GP
a1 = first term of the GP
r = common ratio of the GP
In this case, the first term of the GP is 4:
a1 = 4
Now, let's find the sum of the first 10 terms for both the AP and GP to calculate the difference.
Sum of AP:
The sum of the first n terms of an AP can be calculated using the following formula:
Sn = (n/2) * (2a1 + (n-1) * d)
We want to find the sum of the first 10 terms of the AP:
n = 10
a1 = -2 (as given in the question)
Using the formula:
S10 = (10/2) * (-4 + (10 - 1) * d)
Simplifying further:
S10 = 5 * (-4 + 9d)
We cannot calculate the sum of the AP without knowing the common difference "d."
Sum of GP:
The sum of the first n terms of a GP can be calculated using the following formula:
Sn = a1 * (1 - r^n) / (1 - r)
We want to find the sum of the first 10 terms of the GP:
n = 10
a1 = 4 (as given in the question)
Using the formula:
S10 = 4 * (1 - r^10) / (1 - r)
We cannot calculate the sum of the GP without knowing the common ratio "r."
In conclusion, we are unable to find the exact difference between the sum of the tenth terms of the AP and GP without knowing the common difference or common ratio respectively.