Question

the sum, ( s_n ), of the first ( n ) terms of an arithmetic sequence is given by ( s_n = \frac{n}{2}(a_1 + a_n) ), in which ( a_1 ) is the first term and ( a_n ) is the ( n )th term. the sum ( s_n ), of the first ( n ) terms of a geometric sequence is given by ( s_n = \frac{a_1(1 - r^n)}{1 - r} ), in which ( a_1 ) is the first term and ( r ) is the common ratio (( r
eq 1 )).
use the appropriate formula shown above to find ( 1 + 2 + 3 + 4 + dots + 128 ), the sum of the first 128 natural numbers.
( s_{128} = square )

Explanation:

Step1: Identify the values of \(n\), \(a_1\), and \(a_n\)

For the sum of the first \(n\) natural numbers, \(n = 128\), \(a_1=1\), and \(a_{n}=128\).

Step2: Substitute into the arithmetic - sequence sum formula

The formula for the sum of the first \(n\) terms of an arithmetic sequence is \(S_{n}=\frac{n(a_{1} + a_{n})}{2}\).
Substitute \(n = 128\), \(a_1 = 1\), and \(a_{n}=128\) into the formula:
\(S_{128}=\frac{128\times(1 + 128)}{2}\)
First, calculate the sum inside the parentheses: \(1+128 = 129\).
Then, calculate \(\frac{128\times129}{2}\). Since \(128\div2=64\), we have \(S_{128}=64\times129\).
\(64\times129=(60 + 4)\times129=60\times129+4\times129\)
\(60\times129 = 7740\) and \(4\times129 = 516\)
\(7740+516=8256\)

Answer:

\(8256\)