Calculate the 8th term of the arithmetic progression: 3, 8, 13, 18, ...
Determine the sum of the first 10 terms of the arithmetic progression: 2, 6, 10, 14, ...
Find the common difference of the arithmetic progression: 5, 11, 17, 23, ...
Calculate the 15th term of the arithmetic progression: 4, 9, 14, 19, ...
Solve the following system of linear equations: 2x + 3y = 5 and 4x - y = 3.
Find the value of x and y: 3x + 4y = 7 and x - 2y = 3.
Solve for x and y: 5x + 2y = 10 and 3x - y = 4.
Determine the solution for the system: x + y = 6 and 2x - 3y = 4.
Solve the following system: 2x - y = 1 and 3x + 4y = 12.
Find the values of x and y: x - 3y = 2 and 4x + y = 7.
Solve for x and y: 3x + 2y = 8 and x - y = 1.
Determine the solution for the system: 2x + y = 5 and 3x - 2y = 4.
Solve the following system: x + 2y = 7 and 4x - y = 3.
Find the values of x and y: 2x - 3y = 4 and x + y = 5.
Answer (1)
Let's break down the arithmetic progression problems and solve them one by one:
Calculate the 8th term of the arithmetic progression:
Arithmetic progression: 3 , 8 , 13 , 18 , …
Common difference d = 8 − 3 = 5
First term a 1 = 3
Formula for the n -th term: a n = a 1 + ( n − 1 ) d
Substitute values for the 8th term: a 8 = 3 + ( 8 − 1 ) × 5 = 3 + 35 = 38
Therefore, the 8th term is 38 .
Determine the sum of the first 10 terms of the arithmetic progression:
Arithmetic progression: 2 , 6 , 10 , 14 , …
Common difference d = 6 − 2 = 4
First term a 1 = 2
Formula for the sum of the first n terms: S n = 2 n ( a 1 + a n )
Find the 10th term a 10 = 2 + ( 10 − 1 ) × 4 = 2 + 36 = 38
Substitute values for the sum of the first 10 terms: S 10 = 2 10 ( 2 + 38 ) = 5 × 40 = 200
Therefore, the sum of the first 10 terms is 200 .
Find the common difference of the arithmetic progression:
Arithmetic progression: 5 , 11 , 17 , 23 , …
The common difference d = 11 − 5 = 6
Therefore, the common difference is 6 .
Calculate the 15th term of the arithmetic progression:
Arithmetic progression: 4 , 9 , 14 , 19 , …
Common difference d = 9 − 4 = 5
First term a 1 = 4
Substitute values for the 15th term: a 15 = 4 + ( 15 − 1 ) × 5 = 4 + 70 = 74
Therefore, the 15th term is 74 .
