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If the sum of the first 30 positive odd integers is k, what is the sum of first 30 non-negative even integers?
- a)k-29
- b)k-30
- c)k
- d)k+29
- e)k+30
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
If the sum of the first 30 positive odd integers is k, what is the sum...
Given
- 1 + 3 + 5………..30*2 -1 = k.
- Let’s call this sequence O
To Find: 0 + 2+ 4………30 *2 -2 = ?
- Let’s call this sequence E
Approach
- To express the sum of sequence E in terms of k, we need to express the terms of sequence E in terms of sequence O
- Now, we see that we can write 0 = 1 – 1
- Similarly, we can write 2 = 3 -1
- Continuing the same pattern, we can write 58 = 59 -1
- Observe that 1, 3…..59 are terms of sequence O. So, using the above process we have captured all the terms of sequence O in sequence E
- We will use the above logic to represent the sum of sequence E in terms of k
Working Out
- 0 + 2+ 4……58 = (1-1) + (3-1) +…….(59- 1) = 1+ 3+ 5…….59 – (1 + 1 + 1……….30 times)
- 0 + 2 + 4 …….+ 58 = k – 30
Answer: B
Most Upvoted Answer
If the sum of the first 30 positive odd integers is k, what is the sum...
To find the sum of the first 30 non-negative even integers, we need to consider that the first even integer is 0, the second even integer is 2, the third even integer is 4, and so on.
We can see that the sum of the first n even integers is given by the formula:
Sum = n * (n-1)
Thus, the sum of the first 30 non-negative even integers is:
Sum = 30 * (30-1)
Simplifying this expression, we have:
Sum = 30 * 29
Now, let's compare this sum with the given sum of the first 30 positive odd integers, which is denoted as k.
In order to find the sum of the first 30 positive odd integers, we need to consider that the first odd integer is 1, the second odd integer is 3, the third odd integer is 5, and so on.
We can see that the sum of the first n positive odd integers is given by the formula:
Sum = n^2
Thus, the sum of the first 30 positive odd integers is:
k = 30^2
Simplifying this expression, we have:
k = 900
Now, we can substitute this value of k into the expression for the sum of the first 30 non-negative even integers:
Sum = 30 * 29
Substituting the values, we get:
Sum = 870
So, the sum of the first 30 non-negative even integers is 870, which matches option B, k-30.
We can see that the sum of the first n even integers is given by the formula:
Sum = n * (n-1)
Thus, the sum of the first 30 non-negative even integers is:
Sum = 30 * (30-1)
Simplifying this expression, we have:
Sum = 30 * 29
Now, let's compare this sum with the given sum of the first 30 positive odd integers, which is denoted as k.
In order to find the sum of the first 30 positive odd integers, we need to consider that the first odd integer is 1, the second odd integer is 3, the third odd integer is 5, and so on.
We can see that the sum of the first n positive odd integers is given by the formula:
Sum = n^2
Thus, the sum of the first 30 positive odd integers is:
k = 30^2
Simplifying this expression, we have:
k = 900
Now, we can substitute this value of k into the expression for the sum of the first 30 non-negative even integers:
Sum = 30 * 29
Substituting the values, we get:
Sum = 870
So, the sum of the first 30 non-negative even integers is 870, which matches option B, k-30.
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