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The sum of all terms of the arithmetic progression having ten terms except for the first term, is 99, and except for the sixth term, is 89. Find the 8th term of the progression if the sum of the first and the fifth term is equal to 10.
- a)15
- b)25
- c)18
- d)10
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
The sum of all terms of the arithmetic progression having ten terms ex...
To solve this problem, we can start by setting up the arithmetic progression. Let's assume the first term is 'a' and the common difference is 'd'.
Setting up the arithmetic progression:
The sum of all terms except the first term is 99. Since there are 10 terms, the sum of the remaining 9 terms can be expressed as (9/2)(2a + (9-1)d) = 99.
The sum of all terms except the sixth term is 89. Since there are 10 terms, the sum of the remaining 9 terms can be expressed as (9/2)(2a + (9-1)d) - (a + 5d) = 89.
Now, let's solve these two equations simultaneously to find the values of 'a' and 'd'.
(9/2)(2a + 8d) = 99 ...(1)
(9/2)(2a + 8d) - (a + 5d) = 89 ...(2)
Simplifying equation (2):
(9/2)(2a + 8d) - (a + 5d) = 89
9(2a + 8d) - 2(a + 5d) = 178
18a + 72d - 2a - 10d = 178
16a + 62d = 178 ...(3)
Substituting equation (1) into equation (3):
16a + 62d = 178
16(99/9) + 62d = 178
(16*11) + 62d = 178
176 + 62d = 178
62d = 178 - 176
62d = 2
d = 2/62
d = 1/31
Now, substitute the value of 'd' into equation (1) to find 'a':
(9/2)(2a + 8(1/31)) = 99
(9/2)(2a + 8/31) = 99
(9/2)(62a + 8)/31 = 99
(558a + 72)/31 = 99
558a + 72 = 99 * 31
558a + 72 = 3069
558a = 3069 - 72
558a = 2997
a = 2997/558
a = 27/5
Now that we know the values of 'a' and 'd', we can find the 8th term of the arithmetic progression.
8th term = a + 7d
8th term = (27/5) + 7(1/31)
8th term = (27/5) + (7/31)
8th term = (27*31 + 5*7)/(5*31)
8th term = (837 + 35)/(155)
8th term = 872/155
8th term = 5.632
Therefore, the 8th term of the arithmetic progression is approximately 5.632.
Setting up the arithmetic progression:
The sum of all terms except the first term is 99. Since there are 10 terms, the sum of the remaining 9 terms can be expressed as (9/2)(2a + (9-1)d) = 99.
The sum of all terms except the sixth term is 89. Since there are 10 terms, the sum of the remaining 9 terms can be expressed as (9/2)(2a + (9-1)d) - (a + 5d) = 89.
Now, let's solve these two equations simultaneously to find the values of 'a' and 'd'.
(9/2)(2a + 8d) = 99 ...(1)
(9/2)(2a + 8d) - (a + 5d) = 89 ...(2)
Simplifying equation (2):
(9/2)(2a + 8d) - (a + 5d) = 89
9(2a + 8d) - 2(a + 5d) = 178
18a + 72d - 2a - 10d = 178
16a + 62d = 178 ...(3)
Substituting equation (1) into equation (3):
16a + 62d = 178
16(99/9) + 62d = 178
(16*11) + 62d = 178
176 + 62d = 178
62d = 178 - 176
62d = 2
d = 2/62
d = 1/31
Now, substitute the value of 'd' into equation (1) to find 'a':
(9/2)(2a + 8(1/31)) = 99
(9/2)(2a + 8/31) = 99
(9/2)(62a + 8)/31 = 99
(558a + 72)/31 = 99
558a + 72 = 99 * 31
558a + 72 = 3069
558a = 3069 - 72
558a = 2997
a = 2997/558
a = 27/5
Now that we know the values of 'a' and 'd', we can find the 8th term of the arithmetic progression.
8th term = a + 7d
8th term = (27/5) + 7(1/31)
8th term = (27/5) + (7/31)
8th term = (27*31 + 5*7)/(5*31)
8th term = (837 + 35)/(155)
8th term = 872/155
8th term = 5.632
Therefore, the 8th term of the arithmetic progression is approximately 5.632.
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Community Answer
The sum of all terms of the arithmetic progression having ten terms ex...
According to the question, we have
a2 + ....... + a10 = 99 ... (i)
and a1 + ....... + a5 + a7 +..... + a10 = 89 ... (ii)
Subtracting (ii) from (i), we get
⇒ a6 – a1 = 10 ⇒ a1 + 5d – a1 = 10
⇒ 5d = 10 ⇒ d = 2
Also, a1 + a5 = 10 ⇒ a1 + a1 + 4d = 10
⇒ 2a1 + 8 = 10 ⇒ a1 = 1
∴ 8th term = a1 + 7d = 1 + 14 = 15
a2 + ....... + a10 = 99 ... (i)
and a1 + ....... + a5 + a7 +..... + a10 = 89 ... (ii)
Subtracting (ii) from (i), we get
⇒ a6 – a1 = 10 ⇒ a1 + 5d – a1 = 10
⇒ 5d = 10 ⇒ d = 2
Also, a1 + a5 = 10 ⇒ a1 + a1 + 4d = 10
⇒ 2a1 + 8 = 10 ⇒ a1 = 1
∴ 8th term = a1 + 7d = 1 + 14 = 15
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The sum of all terms of the arithmetic progression having ten terms except for the first term, is 99, and except for the sixth term, is 89. Find the 8th term of the progression if the sum of the first and the fifth term is equal to 10.a)15b)25c)18d)10Correct answer is option 'A'. Can you explain this answer?
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