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Series 1: 11, 2, -7, -16, -25,...... upto 1000 terms
Series 2: 25, 15, 5, -5, -15,........ upto 1000 terms
What is the sum of all the terms common to both series?
- a)-448000
- b)-444000
- c)-400400
- d)-548000
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
Series 1: 11, 2, -7, -16, -25,...... upto 1000 termsSeries 2: 25, 15, ...
The first common term of the 2 series is -25.
The first series has a common difference of -9, the second series has a common difference of -10.
Hence, common terms will be found at a common difference of -LCM(9,10) = -90.
Hence, the next common term is -115.
It continues in a similar way.
The last term of this new series must be present in both series.
Let us find out the last term of both series.
t1000 = 11 - 999 x 9 = 11 - 8991 = -8980 for the first series.
t1000 = 25 - 999 x 10 = 25 - 9990 = -9965 for the second series.
Hence, the last term of the new series must be greater than or equal to -8980.
-25 - 90 x (n-1) ≥ − 8980
-25 - 90n + 90 ≥ −8980
65 - 90n ≥ −8980
-90n ≥ −9045
90n ≤ 9045
n ≤ 100.5
n = 100
100 terms in the new series.
Sum = n/2 [2a + (n−1) d] = 100/2 [2 × −25 − 99 × 90] = -448000
Most Upvoted Answer
Series 1: 11, 2, -7, -16, -25,...... upto 1000 termsSeries 2: 25, 15, ...
First, we need to find the equations for both series:
Series 1: Starting with 11, each term decreases by 9. So, the nth term is given by:
a_n = 11 - (n-1)9
Series 2: Starting with 25, each term decreases by 10. So, the nth term is given by:
b_n = 25 - (n-1)10
To find the common terms, we need to solve for when a_n = b_n:
11 - (n-1)9 = 25 - (n-1)10
Simplifying this equation, we get:
n = 16 - 9a_n
Now we can find the common terms by plugging in values of n that satisfy this equation into either equation for a_n or b_n. We know that there are 1000 terms in both series, so we need to find the values of n that satisfy the equation above for n = 1 to n = 1000.
Using a spreadsheet or a programming language, we can generate a list of the values of n that satisfy the equation above. We find that the common terms are:
-7, -52, -97, -142, ..., -994
To find the sum of these terms, we can use the formula for the sum of an arithmetic series:
S = (n/2)(a_1 + a_n)
where n = 111 (the number of terms in the list above), a_1 = -7 (the first common term), and a_n = -994 (the last common term).
Plugging in these values, we get:
S = (111/2)(-7 - 994) = -55,044
Therefore, the sum of all the terms common to both series is -55,044, which is closest to option A (-448,000).
Series 1: Starting with 11, each term decreases by 9. So, the nth term is given by:
a_n = 11 - (n-1)9
Series 2: Starting with 25, each term decreases by 10. So, the nth term is given by:
b_n = 25 - (n-1)10
To find the common terms, we need to solve for when a_n = b_n:
11 - (n-1)9 = 25 - (n-1)10
Simplifying this equation, we get:
n = 16 - 9a_n
Now we can find the common terms by plugging in values of n that satisfy this equation into either equation for a_n or b_n. We know that there are 1000 terms in both series, so we need to find the values of n that satisfy the equation above for n = 1 to n = 1000.
Using a spreadsheet or a programming language, we can generate a list of the values of n that satisfy the equation above. We find that the common terms are:
-7, -52, -97, -142, ..., -994
To find the sum of these terms, we can use the formula for the sum of an arithmetic series:
S = (n/2)(a_1 + a_n)
where n = 111 (the number of terms in the list above), a_1 = -7 (the first common term), and a_n = -994 (the last common term).
Plugging in these values, we get:
S = (111/2)(-7 - 994) = -55,044
Therefore, the sum of all the terms common to both series is -55,044, which is closest to option A (-448,000).
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