The sum of n terms of the sequence 3,9,15,21 is 7500.Determine the value of n.
Sn=½n(2a+(n-1)d)
Or
Sn=½n[(a+(a+(n-1)d]
Remember that
Tn=a+(n-1)d
a=3 d=6
Tn=3+(n-1)6
Tn=3+6n-6
Tn=(6n-3)
But sn=½n(a+tn)=7500
½n(3+6n-3)=7500
½n(6n)=7500
3n²=7500
n²=2500
n²=50²
n=50
it is correct
Well, math problems can be a little dry sometimes. Let's add a little humor to this one, shall we?
To find the value of n, let's first look at the sequence. 3, 9, 15, 21... it seems to be increasing by 6 each time.
So, let's say the first term of the sequence is "just another manic Monday". And the common difference between terms is "caffeine" because that seems to make everything go faster, right?
Now, we want the sum of n terms to be 7500. Let's see if we can juggle some numbers to figure out this value of n.
The formula for the sum of an arithmetic series is S = (n/2) * (2a + (n-1)d), where S is the sum, a is the first term, and d is the common difference.
We know the sum is 7500, the first term is 3, and the common difference is 6 (because that's how much brighter I seem to shine).
Let's plug these values into the formula and see what we get!
7500 = (n/2) * (2*3 + (n-1)*6)
Now, let's simplify this equation and solve for n.
7500 = (n/2) * (6 + 6n - 6)
7500 = (n/2) * (6n)
Divide both sides by 6 to make Clowntastic progress:
1250 = n^2
Taking the square root of both sides (with a very silly hat on top):
√1250 = n
And after some much clownfusing calculations, we find:
n ≈ 35.355
So, the value of n is approximately 35.355. But, since n has to be a whole number, let's round down to the nearest whole number.
Therefore, the value of n is approximately 35.
And there you have it! The value of n is 35, just like the number of cookie crumbs I leave behind after a snack.
To determine the value of n, we need to set up an equation using the formula for the sum of an arithmetic series.
The formula for the sum of an arithmetic series is:
S = (n/2)(2a + (n-1)d)
Where:
S is the sum of the series,
n is the number of terms,
a is the first term of the series, and
d is the common difference.
In this case, the first term (a) is 3, the common difference (d) is 9 - 3 = 6, and the sum of the series (S) is 7500.
7500 = (n/2)(2*3 + (n-1)*6)
Simplifying the equation:
7500 = 3n + 6(n-1)
7500 = 3n + 6n - 6
7500 = 9n - 6
7500 + 6 = 9n
7506 = 9n
n = 7506/9
n ≈ 834.
To determine the value of n, we need to find the number of terms that sum up to 7500.
Let's observe the given sequence: 3, 9, 15, 21.
We can notice that this is an arithmetic sequence with a common difference of 6 (each term is 6 more than the previous one).
To find the sum of the first n terms of an arithmetic sequence, we can use the formula: Sn = n/2 * (2a + (n-1)d), where Sn is the sum of the first n terms, a is the first term, d is the common difference, and n is the number of terms.
In our case, the first term (a) is 3, the common difference (d) is 6, and the sum of the first n terms (Sn) is 7500.
Plugging these values into the formula, we get:
7500 = n/2 * (2*3 + (n-1)*6)
Simplifying further:
7500 = n/2 * (6 + 6n - 6)
7500 = n/2 * (6n)
Now, we can solve this equation for n.
Dividing both sides by n/2 and simplifying:
7500 = 3n^2
2500 = n^2
n = √2500
n = 50
Therefore, the value of n is 50.