An A. P consists of 21 terms. The sum of three terms in the middle is 129 and the sum of the last three terms is 429. Find the A. P.
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Correct Question :
An A. P consists of 21 terms. The sum of three terms in the middle is 129 and the sum of the last three terms is 237. Find the A. P.
AnswEr :
- This Arithmetic Progression consists of 21 terms.
- Sum of three middle terms are 129
- Sum of last three terms are 237
- Find the AP.?
Let the First term of AP be a and the Common Difference be d.
• Taking First Part of the Question :
Sum of three Middle Terms are 129.
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21 term.
We can see clearly that the 10th, 11th and 12th term is Middle Three Terms.
⋆ Nth Term = {a + (n - 1)d}
⇒ (10th + 11th + 12th) term = 129
⇒ (a + 9d) + (a + 10d) + (a + 11d) = 129
⇒ 3a + 30d = 129
⇒ 3(a + 10d) = 129
- Dividing Both term by 3
⇒ a + 10d = 43⠀⠀⠀⠀⠀⠀—eq. ( I )
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• Taking Second Part of the Question Now :
Sum of last three terms are 237.
we can clearly notice that 19th, 20th and 21st is the last three terms.
⇒ (19th + 20th + 21st) term = 237
⇒ (a + 18d) + (a + 19d) + (a + 20d) = 237
⇒ 3a + 57d = 237
⇒ 3(a + 19d) = 237
- Dividing Both term by 3
⇒ a + 19d = 79⠀⠀⠀⠀⠀⠀—eq. ( II )
━━━━━━━━━━━━━━━━━━━━━━━━
• Subtracting eq.( I ) from eq.( II ) :
⇝ a + 19d = 79
⇝ a + 10d = 43
⠀– ⠀–⠀⠀⠀–
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⇝ 19d – 10d = 79 – 43
⇝ 9d = 36
- Dividing Both term by 9
⇝ d = 4 ⠀⠀⠀[ Common Difference ]
• Putting the value of d in eq.( I ) :
⇝ a + 10d = 43
⇝ a + (10 × 4) = 43
⇝ a + 40 = 43
⇝ a = 43 - 40
⇝ a = 3
∴ A.P. will be 3, 7, 11, 15, 19, 23, 27, 31....
Answer:-
The required AP is 3, 7, 11, .........
Step-by-step explanation:
See the attachment for explaination.
