Solving for the 6th Number in a Sequence
The given problem involves determining the 6th number in a sequence of 9 numbers where certain averages of subgroups of these numbers are provided. This type of problem is common in arithmetic and algebra. Let's walk through a step-by-step solution using the given data.
Understanding the Problem
We are given the following information about a sequence of 9 numbers:
The average of all 9 numbers is 30. The average of the first 5 numbers is 25. The average of the last 3 numbers is 35.The key is to find the value of the 6th number in this sequence.
Step-by-Step Solution
Total Sum of 9 Numbers
The average of the 9 numbers is 30.
Equation: [frac{a_1 a_2 a_3 a_4 a_5 a_6 a_7 a_8 a_9}{9} 30]
Calculation: [a_1 a_2 a_3 a_4 a_5 a_6 a_7 a_8 a_9 30 times 9 270]
Total Sum of the First 5 Numbers
The average of the first 5 numbers is 25.
Equation: [frac{a_1 a_2 a_3 a_4 a_5}{5} 25]
Calculation: [a_1 a_2 a_3 a_4 a_5 25 times 5 125]
Total Sum of the Last 3 Numbers
The average of the last 3 numbers is 35.
Equation: [frac{a_7 a_8 a_9}{3} 35]
Calculation: [a_7 a_8 a_9 35 times 3 105]
Calculating the 6th Number
To find the 6th number, we use the total sum of all 9 numbers and subtract the sums of the first 5 and the last 3 numbers.
Calculation: [a_6 270 - 125 - 105 40]
Thus, the 6th number in the sequence is 40.
Verification
To verify our solution, let's check the sum of the numbers:
Total of first 5 numbers 125
Total of last 3 numbers 105
Total of all 9 numbers 270
Total of 6th number 270 - 125 - 105 40
This confirms our solution is correct.
Concluding Remarks
Through systematic calculation and verification, we can find the 6th number in a sequence given certain averages. Understanding how to manipulate and apply the concept of averages and sums is crucial in solving such problems.
Keywords: average, arithmetic sequence, number sequence