Question

find the sum of the first 20 terms of the sequence.
5, 14, 23, 32, ...

$s_{20}=\square$

Explanation:

Step1: Identify the sequence type

The sequence is \(5, 14, 23, 32, \dots\). Let's check the common difference \(d\). \(14 - 5 = 9\), \(23 - 14 = 9\), \(32 - 23 = 9\). So it's an arithmetic sequence with first term \(a_1 = 5\) and common difference \(d = 9\).

Step2: Recall the sum formula for arithmetic sequence

The sum of the first \(n\) terms of an arithmetic sequence is given by \(S_n=\frac{n}{2}[2a_1+(n - 1)d]\) or \(S_n=\frac{n}{2}(a_1 + a_n)\), where \(a_n=a_1+(n - 1)d\).

Step3: Calculate the 20th term (\(a_{20}\))

Using \(a_n=a_1+(n - 1)d\), with \(n = 20\), \(a_1 = 5\), \(d = 9\):
\(a_{20}=5+(20 - 1)\times9=5 + 19\times9=5+171 = 176\)

Step4: Calculate the sum \(S_{20}\)

Using \(S_n=\frac{n}{2}(a_1 + a_n)\), with \(n = 20\), \(a_1 = 5\), \(a_{20}=176\):
\(S_{20}=\frac{20}{2}(5 + 176)=10\times181 = 1810\)

Answer:

\(1810\)