The third term of a linear sequence is 16 and its fifth term is 34. Find the second term.
Answer (1)
Set up equations based on the given information about the third and fifth terms of the linear sequence.
Solve for the common difference a by eliminating b from the equations.
Substitute the value of a back into one of the equations to solve for b .
Find the second term by substituting n = 2 into the linear sequence formula: 7 .
Explanation
Understanding the Problem We are given that the third term of a linear sequence is 16 and the fifth term is 34. We want to find the second term of this sequence. A linear sequence can be represented as a n = an + b , where a is the common difference and b is a constant.
Setting up Equations We have two equations from the given information:
Third term: a 3 = 3 a + b = 16 Fifth term: a 5 = 5 a + b = 34
Eliminating b Subtract the first equation from the second to eliminate b :
( 5 a + b ) − ( 3 a + b ) = 34 − 16 2 a = 18
Solving for a Solve for a :
a = 2 18 = 9
Solving for b Substitute the value of a into the first equation to solve for b :
3 ( 9 ) + b = 16 27 + b = 16
Finding b Solve for b :
b = 16 − 27 = − 11
Defining the Sequence Now we have the linear sequence defined as:
a n = 9 n − 11
Finding the Second Term To find the second term, substitute n = 2 into the equation:
a 2 = 9 ( 2 ) − 11 = 18 − 11 = 7
Final Answer Therefore, the second term of the linear sequence is 7.
Examples
Linear sequences are useful in many real-world scenarios, such as predicting future values based on a pattern. For example, if you save a fixed amount of money each month, the total savings form a linear sequence. Understanding linear sequences helps in financial planning, predicting inventory levels, or even understanding simple growth patterns in nature.
