Question

Find the difference and 201 terms of the arithmetic progression 26, 29, 32

Answer 1

$\displaystyle\bf\\a_{1} =26\\\\a_{2} =29\\\\a_{2} =a_{1} +d\\\\d=a_{2} -a_{1} =29-26=3\\\\\boxed{\boxed{d=3}}\\\\\\a_{201} =a_{1} +200d=26+200\cdot3=26+600=626\\\\\boxed{\boxed{a_{201} =626}}$

Answer 2

An arithmetic progression (AP) is a sequence of numbers in which the difference of any two successive members is a constant. This constant difference is commonly denoted by \(d\).

The general form of an arithmetic progression is given by:

\[ a_n = a_1 + (n-1)d \]

where: - \( a_n \) is the \(n\)-th term, - \( a_1 \) is the first term, - \( d \) is the common difference, and - \( n \) is the term number.

In your case, the given arithmetic progression is 26, 29, 32, ... where \( a_1 = 26 \) and the common difference \( d = 29 - 26 = 3 \).

Now, let's find the difference and the 201st term.

Difference (\(d\)):

The common difference \( d \) is simply the difference between any two successive terms. In this case:

\[ d = 29 - 26 = 3 \]

So, the common difference is 3.

\(n\)-th Term (\(a_{201}\)):

The formula for the \(n\)-th term of an arithmetic progression is:

\[ a_n = a_1 + (n-1)d \]

We want to find the 201st term (\(a_{201}\)), so substitute \( n = 201 \), \( a_1 = 26 \), and \( d = 3 \) into the formula:

\[ a_{201} = 26 + (201-1) \times 3 \]

\[ a_{201} = 26 + 200 \times 3 \]

\[ a_{201} = 26 + 600 \]

\[ a_{201} = 626 \]

So, the 201st term of the arithmetic progression is 626.

In summary: - The common difference (\(d\)) is 3. - The 201st term (\(a_{201}\)) is 626.