What is the sum of the first 25 terms of an arithmetic sequence if the sum of its 8th and 18th term is 72?
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The sum of the first n terms of an arithmetic sequence S_{n} = \n \over 2 \\) \\ \left\lbrack 2a_1 + \lbrack n-1 \rbrack d \right\rbrack \\)
Where a_{1} is the first term, 'n' is the number of terms and 'd' is the common difference.
To find the sum of the first 25 terms of the arithmetic sequence, you need the first term t_{1} and the common difference 'd'.
Data given: Sum of 8th term and 18th term of the sequence is 72.
The 8th term of the arithmetic progression, t_{8} = t_{1} + (8 - 1)d
or t_{8} = t_{1} + 7d
The 18th term of the sequence t_{18} = t_{1} + (18 - 1)d
or t_{18} t_{1} + 17d
Sum of the 8th and 18th term is 72
t_{8} + t_{18} = t_{1} + 7d + t_{1} + 17d = 72
or 2(t_{1}) + 24d = 72 .... (eqn 1)
The sum of the first n terms of an arithmetic sequence S_{n} = \n \over 2 \\) \\ \left\lbrack 2t_1 + \lbrack n-1 \rbrack d \right\rbrack \\)
The sum of the first 25 terms of an arithmetic sequence S_{25} = \25 \over 2 \\) \\ \left\lbrack 2t_1 + \lbrack 25-1 \rbrack d \right\rbrack \\)
S_{25} = \25 \over 2 \\) \\ \left\lbrack 2t_1 + 24d \right\rbrack \\) ... (eqn 2)
The missing information is the value of \\ \left\lbrack 2t_1 + 24d \right\rbrack \\)
In Step 1 in eqn (1), we found the value of \\ \left\lbrack 2t_1 + 24d \right\rbrack \\) as 72
Substituting the value of 2t_{1} + 24d as 72 in eqn (2), we get S_{25} = \25 \over 2 \\) \\ \left\lbrack 72 \right\rbrack \\)
S_{25} = 25 * 36 = 900
Therefore, sum of the first 25 terms S_{25} = 25 * t_{13}.
We will have the answer when we compute the value of t_{13}
In an AP, the value of any term of the sequence is the arithmetic mean of two terms equidistant from it.
For instance, t_{2} = \t_1 + t_3 \over 2 \\) and t_{5} = \t_2 + t_8 \over 2 \\)
So, t_{13} = \t_8 + t_{18} \over 2 \\)
We know that t_{8} + t_{18} = 72
Hence, t_{13} = \t_8 + t_{18} \over 2 \\) = \72 \over 2 \\) = 36
Sum of the first 25 terms, S_{25} = 25 * 36
S_{25} = 900
Get a quick overview on the basic concepts of Arithmetic sequence including formulae to find the nth term of an arithmetic sequence and the sum of n terms of an arithmetic sequence.
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