SOLUTION: The first term of an AP is 7 and the sum of the first 40 terms is 1216. find the nth term and 20th term

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Question 1203542: The first term of an AP is 7 and the sum of the first 40 terms is 1216.
find the nth term and 20th term
Found 3 solutions by ikleyn, josgarithmetic, Theo:
Answer by ikleyn(52903)   (Show Source):
You can put this solution on YOUR website!
.
The first term of an AP is 7 and the sum of the first 40 terms is 1216
find the nth term and 20th term
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The formula for the sum of the first n terms of any AP is = , where is the 1st term and d is the common difference. Substituting the given values = 7 and = 1216, n= 40 into the formula, you get this equation for "d" 1216 = . It gives you 1216 - 7*40 = 936 = 936/39 = 20*d 24 = 20*d d = 24/20 = 1.2. So, the common difference is d= 1.2. Now, the n-th term is = = 7 + (n-1)*1.2. In particular, = 7 + (20-1)*1.2 = 7 + 19*1.2 = 29.8. ANSWER. The n-th term is = 7 + (n-1)*1.2. The 20th term is 29.8.

Solved.

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For introductory lessons on arithmetic progressions see
    - Arithmetic progressions
    - The proofs of the formulas for arithmetic progressions
    - Problems on arithmetic progressions
    - Word problems on arithmetic progressions
in this site.

Answer by josgarithmetic(39630)   (Show Source):
You can put this solution on YOUR website!
sum of arithmetic sequence for n terms, ;
the general term, for common difference d and term index n.

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first term of an AP is 7 and the sum of the first 40 terms is 1216.
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With this, find the common difference, d,... and then,......

Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website!
here's a reference.
https://byjus.com/maths/arithmetic-progression/

the formulas are:
an = a + (n-1) * d
sn = n/2 * (2a + (n-1) * d)
sn = n/2 * (a + l)

an is the nth term
a is the first term, also able to be called a1.
n is the number of terms in the progression.
d is the common difference between the value of a term and the value of the term that comes next in the progression.
sn is the nth term in the progression.
l is the last term in the progression.

as example, let n = 5 and d = 2 and a = 3

a5 = a + (n-1) * 2 becomes a5 = 3 + 4 * 2 which becomes 11.
the progression will be 3,5,7,9,11, confirming the formula is dorrect.

s5 = n/2 * (2a + (n-1) * d) becomes s5 = 5/2 * (2 * 3 + 4 * 2) which becomes 5/2 * (6 + 8) which becomes 5/2 * 14 which becomes 35.
the sum of the progression is 3 + 5 + 7 + 9 + 11 which is equal to 35, confirming the formula is correct.

the other formula for sn is s5 = n/2 * (a + l) which becomes 5/2 * (3 + 11) which becomes 5/2 * 14 which becomes 35, confirming the formula is correct.

in your problem, you have:

a = 7
s40 = 1216

you want to find an and a20.
n is equal to 40, as given in the description of the problem.

start with the sn formula, which is sn = n/2 * (2a + (n-1) * d)
since n = 40 and a = 7 and sn = 1216, the formula becomes 1216 = 20 * (14 + 39 * d)
simplify to get 1216 = 280 + 780 * d
subtract 280 from both sides of the equation to get 936 = 780 * d
solve for d to get d = 936 / 780 = 1.2

you now have:
a = 7
n = 40
d = 1.2
sn = 1216

you can now solve for an.
formula is an = a + (n-1) * d which becomes a40 = 7 + 39 * 1.2 which becomes a40 = 53.8.

your alternate formula for sn is sn = n/2 * (a + l) which becomes sn = 40/2 * (7 + 53.8) which becomes sn = 20 * 60.8 which becomes 1216.
since this is the same as given, this confirms that value of l is 53.8 and the formula of sn = n/2 * (a + l) is correct.

finally, you are asked to the find the value of the 20th term.
an = a + (n-1) * d becomes a20 = 7 + 19 * 1.2 which becomes 29.8.

as a bonus, the sum of the first 20 terms will use the formula of s20 = 20/2 * (7 + 29.8) which is equal to 368.

i used excel to confirm all these values are correct.
here's the excel worksheet display.