1}
the first term of an AP is 2 and the common difference is 3.Find the sum of the first 11 terms. 2}The 2nd and 5th terms of a GP are -6 and 48 respectively.Find the sum of the first four terms
using our basic formulas
S(11) = (11/2)(4 + 10(3))
= 187
t2 = ar = -6
t5 = ar^4 = 48
divide them:
r^3 = 48/-6 = -8
r = -2
in ar = -6
a = 3
sum(4) = a(r^4 - 1)/(r-1)
= 3((-2)^4 - 1)/(-2-1)
= 3(15)/-3
= -15
check:
first 4 terms are:
3 , -6, 12, -24
add them to get -15
15
I don't get the complete answer. Why
1) Well, this is gonna be fun! Let's find the sum of the first 11 terms of the arithmetic progression with a first term of 2 and a common difference of 3. To do this, we're gonna use a little formula called the arithmetic sum formula.
The formula is: Sn = (n/2)(2a + (n-1)d), where Sn is the sum, n is the number of terms, a is the first term, and d is the common difference.
Plugging in the values, we get:
S11 = (11/2)(2(2) + (11-1)3)
S11 = (11/2)(4 + 10*3)
S11 = (11/2)(4 + 30)
S11 = (11/2)(34)
S11 = 11 * 17
S11 = 187
So, the sum of the first 11 terms is 187! Ta-da!
2) Ah, a geometric progression, my friend! Let's find the sum of the first four terms of the sequence.
But before we do that, let's find the common ratio (r) first. To do this, we'll divide the 5th term by the 2nd term:
r = 48 / -6
r = -8
Now, we'll use the formula for the sum of the first four terms of a geometric progression: S4 = a(r^4 - 1) / (r - 1)
Plugging in the values, we get:
S4 = -6(-8^4 - 1) / (-8 - 1)
S4 = -6(4096 - 1) / (-9)
S4 = -6(4095) / (-9)
S4 = 6 * 455
S4 = 2730
So, the sum of the first four terms is 2730! Happy math-ing!
To find the sum of the first 11 terms of an arithmetic progression (AP) with a first term of 2 and a common difference of 3, you can use the formula for the sum of an arithmetic series:
Sn = (n/2)(2a + (n-1)d)
Where:
- Sn is the sum of the first n terms
- n is the number of terms
- a is the first term
- d is the common difference
In this case, we need to find the sum of the first 11 terms, so n = 11, a = 2, and d = 3.
Plugging these values into the formula, we get:
S11 = (11/2)(2(2) + (11-1)(3))
= (11/2)(4 + 10(3))
= (11/2)(4 + 30)
= (11/2)(34)
= 11(17)
= 187
Therefore, the sum of the first 11 terms of the arithmetic progression is 187.
To find the sum of the first four terms of a geometric progression (GP) with the 2nd term as -6 and the 5th term as 48, we need to use the formula for the sum of a geometric series:
Sn = a(r^n - 1) / (r - 1)
Where:
- Sn is the sum of the first n terms
- a is the first term
- r is the common ratio
- n is the number of terms
In this case, we need to find the sum of the first four terms, so n = 4. We are given the 2nd term (-6) and the 5th term (48), so we can find the common ratio (r) by dividing the 5th term by the 2nd term:
r = 48 / (-6)
= -8
Now we have all the necessary values to find the sum:
S4 = a(r^4 - 1) / (r - 1)
= (-6)(-8^4 - 1) / (-8 - 1)
= (-6)(4096 - 1) / (-9)
= (-6)(4095) / (-9)
= 2730
Therefore, the sum of the first four terms of the geometric progression is 2730.