Answer: 1300
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Step-by-step explanation:
We start off with 16 seats in the first row. Then we have 16+3 = 19 seats in the next row and 19+3 = 22 seats in the third row, and so on.
The pattern is this: 16, 19, 22, ...
We add on 3 each time we need to get another row of seats. We keep doing this until we have 25 rows in all.
We need to determine how many seats are in row n, where n is some positive whole number, aka n is a natural number.
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Going back to the sequence {16,19,22,...} we can see that this is arithmetic because we have the same gap between the adjacent numbers. That gap being 3. This is the common difference, so d = 3.
The start term is a = 16
The nth term is...
f(n) = a + d(n-1)
f(n) = 16 + 3(n-1)
f(n) = 16 + 3n - 3
f(n) = 3n + 13
This function f(n) will quickly tells us how many seats are in a given row.
For instance, if n = 3, then,
f(n) = 3n + 13
f(3) = 3*3 + 13
f(3) = 9 + 13
f(3) = 22
This matches from the third term in {16, 19, 22, ...}. I recommend you try other values of n to help show why that function works the way it does.
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We did all that work to find the nth term f(n) so that we can then use the formula below to sum all the terms from 1 to n
Sn = sum of the first n terms
Sn = first term + second term + ... + nth term
Sn = (n/2)*(first term + nth term)
Sn = (n/2)*( a + f(n) )
Sn = (n/2)*(16 + 3n+13)
Sn = (n/2)*(3n + 29)
Let's try n = 2 and we get
Sn = (n/2)*(3n + 29)
S2 = (2/2)*(3*2 + 29)
S2 = 35
Then note how the first two terms of {16, 19, 22, ...} add to 16+19 = 35 to help confirm that we have the correct Sn formula.
I should point out that this trick only works for arithmetic sequences.
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We have one more step from here. Plug in n = 25 to find the sum of the first 25 terms of the arithmetic sequence {16,19,22,...}
Sn = (n/2)*(3n + 29)
S25 = (25/2)*(3*25 + 29)
S25 = 1300
Therefore, this theater has a total of 1300 seats which is the final answer.
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You could list out all the first 25 terms of {16,19,22,...}, and then add them up, and you should get 1300 as the result. Doing such a check is tedious and often something I recommend you would use computer software for.
I used computer software to get the following
16+19+22+25+28+31+34+37+40+43+46+49+52+55+58+61+64+67+70+73+76+79+82+85+88 = 1300
This confirms the correct final answer. This part is optional in my opinion. It's just there if you're curious. Note how each term in that long sum above is incrementing by 3 each time.