Answer: Option A. -> 253

2|64009( 253
|4
|----------
45|240
|225
|----------
503| 1509
| 1509
|----------
| X
|----------

Therefore \(\sqrt{64009}=253.\)

Answer: Option C. -> -32
 -  The 10th term is  -32

Answer: Option D. -> 127

 -  The 32nd term is 127

The given series is an arithmetic progression with the first term, a = 3 and common difference, d = 4. We need to find the 32nd term of the series.

The formula to find the nth term of an arithmetic progression is given by:

an = a + (n - 1) d

where,

an is the nth term of the AP

a is the first term of the AP

d is the common difference

n is the position of the term to be found

Substituting the given values in the formula, we get:

a32 = 3 + (32 - 1) 4

a32 = 3 + 31 x 4

a32 = 3 + 124

a32 = 127

Therefore, the 32nd term of the given series is 127, which is Option D.

To summarize the solution:

• The given series is an arithmetic progression with a = 3 and d = 4.
• The formula to find the nth term of an AP is an = a + (n - 1) d.
• Substituting the given values in the formula, we get a32 = 3 + (32 - 1) 4.
• Solving the expression, we get a32 = 127.
• Hence, the correct answer is Option D.

Answer: Option B. -> 20th term
To find the position of 98 in the given series 3, 8, 13, ..., we need to first determine the pattern of the series.
The given series is an arithmetic sequence where the common difference is 5. This means that each term in the sequence is obtained by adding 5 to the previous term.
To find the position of 98 in the sequence, we can use the following formula to find the nth term of an arithmetic sequence:
an = a1 + (n - 1)d
where:an = the nth term of the sequencea1 = the first term of the sequenced = the common differencen = the position of the term we want to find
We know that a1 = 3 and d = 5, and we want to find the value of n for which an = 98. So we can rearrange the formula as follows:
n = (an - a1)/d + 1
Substituting the given values, we get:
n = (98 - 3)/5 + 1n = 20
Therefore, the position of 98 in the given series is the 20th term. So the correct option is B.

Answer: Option D. -> 59

 -  The 35th term is 59

To find the 35th term of an arithmetic sequence, we need to use the formula:

an = a1 + (n-1)d

where:

an = the nth term of the sequence

a1 = the first term of the sequence

d = the common difference between consecutive terms

n = the number of the term we want to find

In this case, we are given that the first term, a1, is 8 and the common difference, d, is 1.5. We want to find the 35th term, so we substitute these values into the formula and solve for an:

a35 = 8 + (35-1)1.5

a35 = 8 + 51

a35 = 59

Therefore, the 35th term of the arithmetic sequence with first term 8 and common difference 1.5 is 59.

To summarize, the solution is as follows:

We are given a first term of 8 and a common difference of 1.5.

We use the formula an = a1 + (n-1)d to find the 35th term.

Substituting the given values, we get a35 = 8 + (35-1)1.5.

Simplifying, we get a35 = 8 + 51 = 59.

Therefore, the answer is option D, 59.

Answer: Option C. -> 697
 -  The 100th term is 697

Answer: Option D. -> 30

 -  95 is the 30th term of the series

Explanation:

To find the number of terms in the series 8, 11, 14, ....95, we need to calculate the number of terms between 8 and 95 that follow the pattern 3n + 5.

A series of numbers can be represented in mathematical terms as a sequence of terms. In this case, the terms of the sequence are given by the formula:

t_n = 3n + 5

where t_n represents the nth term of the sequence and n is a positive integer.

To find the number of terms in the sequence, we need to find the value of n for which the nth term (t_n) is 95. Substituting 95 for t_n in the formula, we get:

95 = 3n + 5

Solving for n, we get:

n = (95 - 5)/3 = 30

Since n represents a positive integer, the number of terms in the sequence is equal to 30.

Thus, the correct option is D.

Answer: Option C. -> 14

 -  -54 is the 14th term in the series

To find the number of terms in the series 11, 6, 1,..., -54, we need to determine the common difference (d) and the nth term of the sequence (tn) first. Then, we can use the formula to find the number of terms (n) in the sequence.

• Common difference (d):

The common difference is the difference between any two consecutive terms in an arithmetic sequence. To find the common difference, we can subtract any two consecutive terms:

d = 6 - 11 = -5 = 1 - 6 = -54 - (-5) = -49

The common difference is -5.

• nth term of the sequence (tn):

The nth term of an arithmetic sequence can be found using the formula:

tn = a + (n-1)d

where a is the first term of the sequence and d is the common difference.

In this sequence, a = 11 and d = -5. Thus, the nth term of the sequence is:

tn = 11 + (n-1)(-5) = 11 - 5n + 5 = 16 - 5n

• Number of terms (n):

To find the number of terms in the sequence, we need to determine the value of n such that tn = -54. Substituting tn and solving for n:

16 - 5n = -54

-5n = -70

n = 14

Therefore, there are 14 terms in the sequence.

Answer: Option C (14)

To summarize:

• The common difference is -5.
• The nth term of the sequence is tn = 16 - 5n.
• The number of terms is 14, as tn = -54 when n = 14.

If you think the solution is wrong then please provide your own solution below in the comments section .

Answer: Option A. -> 41

 -  In the series 62 is the 41st term

The given series is an arithmetic progression (A.P.) with the first term 'a' = 2 and the common difference 'd' = 3.5 - 2 = 1.5.

To find the number of terms in the series, we need to determine the last term. We can use the formula for the nth term of an arithmetic progression to find the last term.

The formula for the nth term of an arithmetic progression is given by:

an = a + (n - 1) * d

where an is the nth term of the A.P., a is the first term, d is the common difference, and n is the number of terms.

We need to find the value of n for which the nth term is 62. So, we can write:

62 = 2 + (n - 1) * 1.5

59 = (n - 1) * 1.5

n - 1 = 59 / 1.5

n - 1 = 39.33

n ≈ 40.33

We got a non-integer value of n, which indicates that there are only 40 terms in the series up to 61. The 41st term would exceed 62.

Hence, the correct option is A (41).

To summarize, we used the following concepts/formulas:

• Arithmetic Progression: A sequence of numbers in which the difference between any two consecutive terms is constant.
• Formula for the nth term of an arithmetic progression: an = a + (n - 1) * d
• Calculation of the number of terms in an arithmetic progression using the nth term and the first term.

If you think the solution is wrong then please provide your own solution below in the comments section .

Answer: Option A. -> 2