Math Problem Statement
The number of terms in Arithmetic sequence is even . The sum of odd terms is 24.if the sum of even terms is 30 and the last term exceeds the first 10.5 ,then find the number of terms ,the first term ,the common difference
Solution
Let's break down the information given:
- The number of terms in the arithmetic sequence is even.
- The sum of the odd terms is 24.
- The sum of the even terms is 30.
- The last term exceeds the first term by 10.5.
Let the number of terms in the sequence be . This is because the number of terms is even, so we denote the number of odd terms and even terms separately.
- Let the first term of the arithmetic sequence be .
- Let the common difference be .
The odd terms form their own arithmetic sequence:
- First odd term: ,
- Second odd term: ,
- Third odd term: , and so on.
There are odd terms, and their sum is given as 24. So, the sum of these odd terms is: Simplifying this:
The even terms also form an arithmetic sequence:
- First even term: ,
- Second even term: ,
- Third even term: , and so on.
There are even terms, and their sum is given as 30. So, the sum of these even terms is: Simplifying this:
Finally, we know that the last term exceeds the first term by 10.5. The last term of the sequence (which is the -th term) is given by:
We now have three equations:
Let's solve these equations step by step.The solution to the system of equations is:
- The number of terms is .
- The first term is .
- The common difference is .
Thus, the arithmetic sequence has 8 terms, with the first term being and the common difference being .
Would you like a detailed verification or further explanation on how these values fit the given conditions? Here are 5 related questions:
- How is the sum of the terms in an arithmetic sequence calculated?
- Can you find the explicit form of the sequence using the values of and ?
- What happens to the sequence if the common difference is negative?
- How would you solve this problem if only the sum of all terms were given?
- Can you explore sequences where the common difference is a fraction?
Tip: In any arithmetic sequence, knowing the first term and the common difference allows you to easily construct any term in the sequence!
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Arithmetic Sequences
Summation of Series
System of Equations
Formulas
Sum of an arithmetic sequence: S_n = n/2 * (first_term + last_term)
nth term of an arithmetic sequence: a_n = a + (n-1)d
Sum of odd terms: S_odd = n * (a + (n-1)d)
Sum of even terms: S_even = n * ((a + d) + (n-1)d)
Theorems
Properties of Arithmetic Sequences
Suitable Grade Level
Grades 9-12
Related Recommendation
Arithmetic Sequence: Find First Term, Difference, and General Term with Conditions a2 + a3 = 23, a9 + a10 = 44
Solving Arithmetic Sequence: Sum of First 3 Terms
Solve Arithmetic Sequence Problem: Sum of Odd and Even Terms
Find the 7th Term of an Arithmetic Sequence with x1 = 5 and x4 + x5 = 24
Find the Number of Terms in Arithmetic Sequences