What are the first three terms of an arithmetic series in which a1 =9,an=105, and Sn=741?
done here
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Well, you've really given me a lot to work with here! It's like a puzzle within a puzzle. Let's see if we can put the pieces together.
We know that the first term (a1) is 9 and the nth term (an) is 105. Now, if we could only find the common difference (d), we'd be golden!
Here's where things get a little tricky, though. Instead of giving me the common difference, you've thrown in the total sum of the series (Sn), which is 741. Sneaky, sneaky!
But fear not, my friend! We can still crack this enigma.
We have this formula for the sum of an arithmetic series: Sn = n/2 * (a1 + an)
Since we know Sn = 741, we can rearrange the formula to solve for n:
741 = n/2 * (9 + 105)
Now, let's do some math magic:
741 = n * (9 + 105)/2
741 = n * 114/2
741 = 57n
n = 741/57
n ≈ 13
Great! We now know that the total number of terms in the series is approximately 13.
But we still gotta find those first three terms, so let's divide and conquer!
We know that a1 = 9, so that's one down.
We also know that the common difference is found by taking the difference between any two adjacent terms. Oh look, we have a second term (105) now!
105 - 9 = 96
Voila! The common difference (d) is 96.
So, let's list the first three terms of our arithmetic series:
a1 = 9
a2 = 9 + 96 = 105
a3 = 105 + 96 = 201
Ta-da! The first three terms of the arithmetic series are 9, 105, and 201. I hope this magical journey through arithmetic land put a smile on your face!
To find the first three terms of an arithmetic series, we need to find the common difference (d) first and then calculate the terms.
Given:
a1 = 9 (first term)
an = 105 (last term)
Sn = 741 (sum of the series)
We can use the formula for the sum of an arithmetic series:
Sn = (n/2)(a1 + an)
Let's solve the formula for n first:
Sn = (n/2)(a1 + an)
741 = (n/2)(9 + 105)
741 = (n/2)(114)
741 = 57n
n ≈ 13
Next, we can find the common difference (d) using the formula:
d = (an - a1) / (n - 1)
d = (105 - 9) / (13 - 1)
d = 96 / 12
d = 8
Now that we have the common difference, we can calculate the first three terms:
a1 = 9
a2 = a1 + d = 9 + 8 = 17
a3 = a2 + d = 17 + 8 = 25
Therefore, the first three terms of the arithmetic series are 9, 17, and 25.
To find the first three terms of an arithmetic series, we need to use the given information:
The formula for the nth term (an) of an arithmetic series is: an = a1 + (n - 1)d, where a1 is the first term and d is the common difference.
The formula for the sum of the first n terms (Sn) of an arithmetic series is: Sn = (n/2)(a1 + an).
We are given that a1 = 9, an = 105, and Sn = 741.
First, let's find the common difference (d):
Using the formula for the nth term: an = a1 + (n - 1)d
105 = 9 + (n - 1)d
Subtracting 9 from both sides: 105 - 9 = (n - 1)d
96 = (n - 1)d
Next, let's find the number of terms (n) using the formula for the sum of the first n terms:
Sn = (n/2)(a1 + an)
741 = (n/2)(9 + 105)
Dividing both sides by (9+105)/2, we get:
741 / (9+105)/2 = n
Simplifying, we find that n = 9.
Now we know the common difference (d) is equal to 96 and the number of terms (n) is equal to 9.
To find the first term (a1), we can substitute these values into the formula for the nth term:
105 = a1 + (9 - 1) * 96
105 = a1 + 8 * 96
105 = a1 + 768
Subtracting 768 from both sides, we get:
a1 = 105 - 768 = -663
So, the first term (a1) is -663.
Now that we have a1 and d, we can find the first three terms by using the formula for the nth term (an):
a1 = -663
d = 96
Using the formula: an = a1 + (n - 1)d
For n = 1: a1 = -663 + (1 - 1) * 96 = -663 + 0 = -663
For n = 2: a2 = -663 + (2 - 1) * 96 = -663 + 96 = -567
For n = 3: a3 = -663 + (3 - 1) * 96 = -663 + 192 = -471
Therefore, the first three terms of the arithmetic series are -663, -567, and -471.