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If 8th term of an arithmetic progression is 31 and its 15th term is 16 more than 11th term of the arithmetic progression. What is the first term of that arithmetic progression?
- a)4
- b)3
- c)-4
- d)-3
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
If 8th term of an arithmetic progression is 31 and its 15th term is 16...
Let a be the first term and d be the common difference
a + 7d = 31
a + 14d = a + 10d + 16
a + 14d - a - 10d = 16
⇒ d = 4
∴ a = 31 - 7d = 31 - 28 = 3
a + 7d = 31
a + 14d = a + 10d + 16
a + 14d - a - 10d = 16
⇒ d = 4
∴ a = 31 - 7d = 31 - 28 = 3
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If 8th term of an arithmetic progression is 31 and its 15th term is 16...
Understanding the Problem
We need to find the first term of an arithmetic progression (AP) given that:
- The 8th term (a + 7d) is 31.
- The 15th term is 16 more than the 11th term.
Let’s denote:
- a = first term of the AP
- d = common difference
Step 1: Express the Terms
- 8th term: a + 7d = 31
- 11th term: a + 10d
- 15th term: a + 14d
From the problem, we know:
a + 14d = (a + 10d) + 16
Step 2: Set Up the Equation
Rearranging the equation gives us:
a + 14d - a - 10d = 16
This simplifies to:
4d = 16
Step 3: Solve for d
Dividing both sides by 4:
d = 4
Step 4: Substitute d Back
Now, substitute d back into the equation for the 8th term:
a + 7(4) = 31
a + 28 = 31
Step 5: Solve for a
By isolating a, we find:
a = 31 - 28
a = 3
Conclusion
The first term of the arithmetic progression is:
- Option B: 3.
We need to find the first term of an arithmetic progression (AP) given that:
- The 8th term (a + 7d) is 31.
- The 15th term is 16 more than the 11th term.
Let’s denote:
- a = first term of the AP
- d = common difference
Step 1: Express the Terms
- 8th term: a + 7d = 31
- 11th term: a + 10d
- 15th term: a + 14d
From the problem, we know:
a + 14d = (a + 10d) + 16
Step 2: Set Up the Equation
Rearranging the equation gives us:
a + 14d - a - 10d = 16
This simplifies to:
4d = 16
Step 3: Solve for d
Dividing both sides by 4:
d = 4
Step 4: Substitute d Back
Now, substitute d back into the equation for the 8th term:
a + 7(4) = 31
a + 28 = 31
Step 5: Solve for a
By isolating a, we find:
a = 31 - 28
a = 3
Conclusion
The first term of the arithmetic progression is:
- Option B: 3.
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