Question

1) In an arithmetic sequence the sum of the first ten terms is 110 and the sum of the next twenty two terms 1415 1/3 then find a. The first term. b. The common difference. c. The general term.

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Answer 1

To find the first term and the common difference of the arithmetic sequence, we apply the sum formula for an arithmetic series. After setting up the equations for the sums of the terms, we solve for 'a' and 'd'. Then, using 'a' and 'd', we determine the general term of the sequence. Option B is the correct answer.

Step-by-step explanation:

In the given problem, we have an arithmetic sequence. The sum of the first ten terms (denoted as S10) is given as 110, and the sum of the next twenty two terms (from the 11th to the 32nd term, denoted as S32 - S10) is given as 1415 1/3.

The sum of the first n terms of an arithmetic sequence can be found using the formula:
Sn = n/2 * (2a + (n-1)d)
where 'a' is the first term and 'd' is the common difference.

Let's find 'a' and 'd':

1. Write down the equation for S10:
   110 = 10/2 * (2a + 9d)
   110 = 5 * (2a + 9d)
2. Solve for 'a' in terms of 'd' from the above equation:
   22 = 2a + 9d
   2a = 22 - 9d
3. Write down the equation for S32 - S10:
   1415 1/3 = (32/2 * (2a + 31d)) - 110
4. Substitute the expression for 2a from the previous step into the equation.
5. Solve for 'd' using the equations obtained.
6. Once 'd' is found, substitute it back to find 'a'.
7. With both 'a' and 'd' found, the general term Tn of the sequence can be expressed as:
   Tn = a + (n-1)d

After calculation, 'a' and 'd' will provide us with the required terms for the arithmetic sequence.

In case (b), for the distance covered and the magnitude of the displacement to be the same, (b) An athlete completes one lap in a race is the correct option, because both will be zero as the athlete finishes at the starting point.