A comprehensive tutorial on arithmetic sequences and summation.
An arithmetic sequence is a sequence of numbers where each term is obtained by adding a constant number to the preceding term. This constant is called the common difference.
The nth term \( a_n \) is given by:
\[ a_n = a_1 + (n - 1)d \]
where \( a_1 \) is the first term and \( d \) is the common difference.
The sum of the first n terms \( S_n \) is given by:
\[ S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2} \left[ 2a_1 + (n - 1)d \right] \]
Sequence: 0, 6, 12, 18, 24, ...
Each term is obtained by adding 6 to the preceding term, so it's an arithmetic sequence with common difference \( d = 6 \).
Which term of the arithmetic sequence 2, 5, 8, ... is equal to 227?
First term \( a_1 = 2 \), common difference \( d = 3 \).
Using the nth term formula:
\[ 227 = 2 + (n - 1) \cdot 3 \]
Solve for \( n \):
\[ n - 1 = \frac{227 - 2}{3} = 75 \]
\[ n = 76 \]
The 76th term is 227.
How many consecutive odd integers, starting from 9, must be added to obtain a sum of 15,860?
First term \( a_1 = 9 \), common difference \( d = 2 \).
Using the sum formula:
\[ S_n = \frac{n}{2} \left[ 2 \cdot 9 + (n - 1) \cdot 2 \right] = 15860 \]
Simplify to:
\[ n^2 + 8n - 15860 = 0 \]
Solving the quadratic:
\[ n = 122 \quad \text{or} \quad n = -130 \]
We take the positive solution: 122 consecutive odd numbers must be added.
The sum of the first \( n \) terms of an arithmetic sequence is \( S_n = 2n^2 + 5n \). Find the first 3 terms.
The nth term is given by \( a_n = S_n - S_{n-1} \):
\[ a_n = (2n^2 + 5n) - [2(n-1)^2 + 5(n-1)] \]
Simplify:
\[ a_n = 4n + 3 \]
Now find the first three terms:
\[ a_1 = 4(1) + 3 = 7 \]
\[ a_2 = 4(2) + 3 = 11 \]
\[ a_3 = 4(3) + 3 = 15 \]
The sum of three consecutive terms in an arithmetic sequence is 27, and their product is 585. Find the three numbers.
Let the three terms be \( x \), \( x+d \), \( x+2d \).
Sum: \( x + (x+d) + (x+2d) = 27 \) → \( 3x + 3d = 27 \) → \( x + d = 9 \) (Equation 1)
Product: \( x(x+d)(x+2d) = 585 \) (Equation 2)
From Equation 1: \( x = 9 - d \). Substitute into Equation 2:
\[ (9-d)(9)(9+d) = 585 \]
\[ 81 - d^2 = 65 \]
\[ d^2 = 16 \]
\[ d = \pm 4 \]
Two solutions:
The first three terms of an arithmetic sequence are: \( x \), \( \frac{5x}{4} \), \( \frac{9}{2} \). Find \( x \) and \( d \).
Common difference must be consistent:
\[ \frac{5x}{4} - x = \frac{9}{2} - \frac{5x}{4} \]
Solve for \( x \):
\[ \frac{x}{4} = \frac{18 - 5x}{4} \]
\[ x = 18 - 5x \]
\[ 6x = 18 \]
\[ x = 3 \]
Then common difference:
\[ d = \frac{5x}{4} - x = \frac{15}{4} - 3 = \frac{3}{4} \]