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Answer :
Sure, let's solve these questions step by step:
5) Find the sum of the first 200 natural numbers.
To find the sum of the first 200 natural numbers, we can use this formula for the sum of the first [tex]\( n \)[/tex] natural numbers:
[tex]\[ \text{Sum} = \frac{n(n + 1)}{2} \][/tex]
Here, [tex]\( n = 200 \)[/tex].
[tex]\[ \text{Sum} = \frac{200 \times (200 + 1)}{2} \][/tex]
[tex]\[ \text{Sum} = \frac{200 \times 201}{2} = 20100 \][/tex]
So, the sum of the first 200 natural numbers is 20,100.
6) Check whether 2, 4, 8, 16,...... is an AP. If this is an AP then find common difference [tex]\( d \)[/tex] and write next 3 terms.
To check if the sequence is an Arithmetic Progression (AP), we find the difference between consecutive terms:
- The difference between 4 and 2 is [tex]\( 4 - 2 = 2 \)[/tex]
- The difference between 8 and 4 is [tex]\( 8 - 4 = 4 \)[/tex]
- The difference between 16 and 8 is [tex]\( 16 - 8 = 8 \)[/tex]
Since these differences are not the same ([tex]\( 2, 4, 8 \)[/tex]), the sequence is not an AP.
Since the sequence is a geometric progression, rather than an AP, with a common ratio of 2, the next three terms would be obtained by multiplying the last term by 2 each time:
- The next term after 16 is [tex]\( 16 \times 2 = 32 \)[/tex]
- The next term after 32 is [tex]\( 32 \times 2 = 64 \)[/tex]
- The next term after 64 is [tex]\( 64 \times 2 = 128 \)[/tex]
So, the next three terms are 32, 64, 128.
7) Determine the AP whose 3rd term is 5 and 7th term is
To determine an arithmetic progression (AP), we need to find its first term [tex]\(a\)[/tex] and common difference [tex]\(d\)[/tex] given some terms in the sequence. We know:
- The 3rd term ([tex]\(a + 2d\)[/tex]) is 5.
However, the 7th term wasn't provided in the information, making it unclear, so we cannot find the specific common difference and first term of this AP without knowing one more condition or term.
If we get more information, like the value of the 7th term or another specific term, we could find the whole sequence.
Let me know if you have any more questions or need further details!
5) Find the sum of the first 200 natural numbers.
To find the sum of the first 200 natural numbers, we can use this formula for the sum of the first [tex]\( n \)[/tex] natural numbers:
[tex]\[ \text{Sum} = \frac{n(n + 1)}{2} \][/tex]
Here, [tex]\( n = 200 \)[/tex].
[tex]\[ \text{Sum} = \frac{200 \times (200 + 1)}{2} \][/tex]
[tex]\[ \text{Sum} = \frac{200 \times 201}{2} = 20100 \][/tex]
So, the sum of the first 200 natural numbers is 20,100.
6) Check whether 2, 4, 8, 16,...... is an AP. If this is an AP then find common difference [tex]\( d \)[/tex] and write next 3 terms.
To check if the sequence is an Arithmetic Progression (AP), we find the difference between consecutive terms:
- The difference between 4 and 2 is [tex]\( 4 - 2 = 2 \)[/tex]
- The difference between 8 and 4 is [tex]\( 8 - 4 = 4 \)[/tex]
- The difference between 16 and 8 is [tex]\( 16 - 8 = 8 \)[/tex]
Since these differences are not the same ([tex]\( 2, 4, 8 \)[/tex]), the sequence is not an AP.
Since the sequence is a geometric progression, rather than an AP, with a common ratio of 2, the next three terms would be obtained by multiplying the last term by 2 each time:
- The next term after 16 is [tex]\( 16 \times 2 = 32 \)[/tex]
- The next term after 32 is [tex]\( 32 \times 2 = 64 \)[/tex]
- The next term after 64 is [tex]\( 64 \times 2 = 128 \)[/tex]
So, the next three terms are 32, 64, 128.
7) Determine the AP whose 3rd term is 5 and 7th term is
To determine an arithmetic progression (AP), we need to find its first term [tex]\(a\)[/tex] and common difference [tex]\(d\)[/tex] given some terms in the sequence. We know:
- The 3rd term ([tex]\(a + 2d\)[/tex]) is 5.
However, the 7th term wasn't provided in the information, making it unclear, so we cannot find the specific common difference and first term of this AP without knowing one more condition or term.
If we get more information, like the value of the 7th term or another specific term, we could find the whole sequence.
Let me know if you have any more questions or need further details!
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