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If the sum of the first five terms of an Arithmetic sequence is equal to 120 and the sum of the next five terms of the same Arithmetic Sequence is equal to 245, what is the 4th term of this Sequence?
- a)29
- b)34
- c)81
- d)86
- e)91
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
If the sum of the first five terms of an Arithmetic sequence is equal ...
Given:
- Sum of the first 5 terms of an arithmetic sequence = 120
- Sum of the next 5 terms of the same arithmetic sequence = 245
- Let the first term of this arithmetic sequence be x1 and let the common difference be d.
To Find:
- 4th term of the arithmetic sequence.
- So the 4th term of the sequence will become x1+3d
- So we need to find the value of x1 and d or the value of x1+3d to find the 4th term of the sequence.
Approach:
- We know that the sum of first n terms of the Arithmetic Sequence is given as
where n is the number of terms in the arithmetic sequence.- Using the formula above for the sum of first 5 terms of the sequence, we will get an equation in terms of and common difference d, as we are given the sum of first 5 terms of the sequence.
- We are also given the sum of next 5 terms of the sequence. So, we will be able to calculate the sum of first 10 terms of the sequence.
→ Sum of first 10 terms of sequence = Sum of first 5 terms + sum of next 5 terms. - Using the formula above for the sum of first 10 terms of sequence, we will get another equation in terms of x1 and common difference d.
- Using these two equations in x1 and d, we will be able to calculate the value of x1 and d.
- Knowing the values of x1 and d, we will be able to calculate the fourth term of the sequence, which is equal to x1+3d
Working out:
- Sum of first 5 terms of the arithmetic sequence = 120
- Putting this in formula of sum of first n terms, where n=5 and z=120, we get
- Sum of the next 5 terms of the sequence = 245
- Sum of the first 10 terms of the sequence = Sum of the first five terms + Sum of the next five terms.
- Sum of the first 10 terms of the sequence = 120+245 = 365
- Sum of the first 10 terms of the sequence = 120+245 = 365
- Now, using the formula of the sum of first n terms of an arithmetic sequence, we get
- Solving Equations 1 and 2.
- Multiplying ‘equation 1’ by 2, we have 10x1+20d =240 ...(Equation 3)
Now that we have values of x1 and d. The value of 4th term of the sequence will be
⇒ x1+3(d)=14+3(5)=29
Answer:
- The value of 4th term of the sequence is 29.
- Hence the correct answer is option A
Alternate method
- Let the first term be 'a' and common dfference between any two cosecutive terms be 'd'
Therefore,
- 1st term = a
- 5th term = a + 4d
- 6th term = a + 5d
- 10th term = a + 9d
- Average of first five terms of an arithemetic sequence = (First term + Last term)/2 = (a + a +4d) / 2 = a + 2d
- Sum of first five terms = Average of first five terms * 5 = (a + 2d) * 5 = 120
- a + 2d = 120/5 = 24 ---------------- Eq(1)
- Average of next five terms of the arithemetic sequence = (First term + Last term)/2 = (a+ 5d + a +9d) / 2 = a + 7d
- Sum of five terms = Average of five terms * 5 = (a + 7d) * 5 = 245
- a + 7d = 245/5 = 49---------------- Eq(2)
Solving Eq(1) and (2) we get
- d = 5
- 4th term =
- a + 3d = (a+2d)+ d = 24 + 5 = 29
Correct Answer: Option A
Most Upvoted Answer
If the sum of the first five terms of an Arithmetic sequence is equal ...
Solution:
Let's assume that the first term of the Arithmetic sequence is 'a' and the common difference is 'd'.
Sum of the first five terms of an Arithmetic sequence can be calculated as:
S1 = 5/2(2a + (5-1)d) = 120
Simplifying the equation, we get:
2a + 4d = 48
a + 2d = 24
Sum of the next five terms of the same Arithmetic Sequence can be calculated as:
S2 = 5/2(2(a+5d) + (5-1)d) = 245
Simplifying the equation, we get:
a + 9d = 49
To find the fourth term of the sequence, we need to calculate 'a+3d'.
We can use both the equations above to solve for 'a' and 'd'.
Solving for 'a' and 'd':
a = 24 - 2d
Substituting the value of 'a' in the second equation:
24 - 2d + 9d = 49
7d = 25
d = 25/7
Substituting the value of 'd' in the first equation:
a + 2d = 24
a + 2(25/7) = 24
a = 24 - (50/7)
a = 10/7
Therefore, the fourth term of the Arithmetic sequence is:
a+3d = (10/7) + 3(25/7) = 95/7 = 13.57
Since the options given are in integers, we need to round off the answer to the nearest integer, which is 14.
Hence, the correct answer is (A) 29.
Let's assume that the first term of the Arithmetic sequence is 'a' and the common difference is 'd'.
Sum of the first five terms of an Arithmetic sequence can be calculated as:
S1 = 5/2(2a + (5-1)d) = 120
Simplifying the equation, we get:
2a + 4d = 48
a + 2d = 24
Sum of the next five terms of the same Arithmetic Sequence can be calculated as:
S2 = 5/2(2(a+5d) + (5-1)d) = 245
Simplifying the equation, we get:
a + 9d = 49
To find the fourth term of the sequence, we need to calculate 'a+3d'.
We can use both the equations above to solve for 'a' and 'd'.
Solving for 'a' and 'd':
a = 24 - 2d
Substituting the value of 'a' in the second equation:
24 - 2d + 9d = 49
7d = 25
d = 25/7
Substituting the value of 'd' in the first equation:
a + 2d = 24
a + 2(25/7) = 24
a = 24 - (50/7)
a = 10/7
Therefore, the fourth term of the Arithmetic sequence is:
a+3d = (10/7) + 3(25/7) = 95/7 = 13.57
Since the options given are in integers, we need to round off the answer to the nearest integer, which is 14.
Hence, the correct answer is (A) 29.
Community Answer
If the sum of the first five terms of an Arithmetic sequence is equal ...
Solve for S5
120 = 5a + 10d ...after solving for S5
Solve for S10 , Here S10 will be First 5 term + further 5 terms , thus S10 = 120+345 = 365
365 = 10a + 45d
Simultaneously solve both the equations and we will get the value of 1st term a = 4 and common difference d = 25/3
Then, solve L4 = a + (n-1)d
= 4 + (4-1)25/3
= 4 + 25
= 29
120 = 5a + 10d ...after solving for S5
Solve for S10 , Here S10 will be First 5 term + further 5 terms , thus S10 = 120+345 = 365
365 = 10a + 45d
Simultaneously solve both the equations and we will get the value of 1st term a = 4 and common difference d = 25/3
Then, solve L4 = a + (n-1)d
= 4 + (4-1)25/3
= 4 + 25
= 29
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If the sum of the first five terms of an Arithmetic sequence is equal to 120 and the sum of the next five terms of the same Arithmetic Sequence is equal to 245, what is the 4th term of this Sequence?a)29b)34c)81d)86e)91Correct answer is option 'A'. Can you explain this answer? for GMAT 2025 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about If the sum of the first five terms of an Arithmetic sequence is equal to 120 and the sum of the next five terms of the same Arithmetic Sequence is equal to 245, what is the 4th term of this Sequence?a)29b)34c)81d)86e)91Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for GMAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the sum of the first five terms of an Arithmetic sequence is equal to 120 and the sum of the next five terms of the same Arithmetic Sequence is equal to 245, what is the 4th term of this Sequence?a)29b)34c)81d)86e)91Correct answer is option 'A'. Can you explain this answer?.
If the sum of the first five terms of an Arithmetic sequence is equal to 120 and the sum of the next five terms of the same Arithmetic Sequence is equal to 245, what is the 4th term of this Sequence?a)29b)34c)81d)86e)91Correct answer is option 'A'. Can you explain this answer? for GMAT 2025 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about If the sum of the first five terms of an Arithmetic sequence is equal to 120 and the sum of the next five terms of the same Arithmetic Sequence is equal to 245, what is the 4th term of this Sequence?a)29b)34c)81d)86e)91Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for GMAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the sum of the first five terms of an Arithmetic sequence is equal to 120 and the sum of the next five terms of the same Arithmetic Sequence is equal to 245, what is the 4th term of this Sequence?a)29b)34c)81d)86e)91Correct answer is option 'A'. Can you explain this answer?.
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