Q. The sum of the first 7 terms of an A.P. is 49 and the sum of the first 17 terms is 289. Find the sum of first n terms.
- Sol. Let the first terms be a and the common difference be d.
- Now, it is given that,
- S_{7} =7/2 {2a + (7 – 1)d} = 49
- ⇒ 49 =7/2 {2a + 6d}
- ⇒ a + 3d = 7 …(i)
- and S_{17}=17/2 {2a + (17 – 1)d} = 289
- ⇒ 289 = 17/2 {2a + (16)d}
- ⇒ a + 8d = 17 …(ii)
- Subtracting equation (i) from equation (ii), we have
- 5d = 10
- ⇒ d = 2
- From equation (i),
- a = 7 – 3d
- = 7 – 3(2)
- = 7 – 6 = 1
- S_{n} =n/2 {2(1) + (n – 1)2}
- ⇒ S_{n} = n/2 {2 + 2n – 2} = n^{2}
- Hence, the sum of n terms will be n^{2}. Ans.
Q. A sum of ₹ 1000 is invested at 8% S.I. per annum. Calculate the rate of interest at the end of 1, 2, 3, … years. Is the sequence of the interests an A.P. ? Find the interest at the end of 30 years.
Ans. The interest after 1 year = ₹ 80
The interest after 2 years = ₹ 160
The interest after 3 years = ₹ 240
Yes, the sequence of the interests is an A.P.
The interest after 30 years = ₹ 2400.
Q. In a flower bed, there are 23 rose plants in the first row, 21 in the second row, 19 in the third row and so on. There are five plants in the last row. How many rows are there in the flower
bed ?
Ans. 10.
Q. A person started working in 1995 at a salary of ₹ 5000 per month with a yearly increment of ₹ 200. In which year did his salary reach ₹ 7000 per month?
- Ans. In the year 2005.
Q. A housewife saved ₹ 5 in the first week of the year and thereafter increased her weekly savings by ₹ 1.75. After how many weeks will her weekly savings be ₹ 20.75?
- Ans. 10 weeks.
- Ans. 198.
- Ans. – 0.5, 2.5.
Q.Find the A.P. whose third term is 16 and the seventh term exceeds its fifth term by 12.
- Ans. 4, 10, 16, 22, …
- Ans. 100.
- Ans. (i) 550, (ii) 775.
Q. The contract of a construction job specifies a penalty for delay of completion beyond a certain date as follows : ₹ 200 for the first day, ₹ 250 for the second day, ₹ 300 for the third day and so on. How much does a delay of 30 days cost the contractor?
- Ans. ₹ 27,750.
Q. A sum of ₹ 280 is to be used to award four prizes. If each prize after the first is ₹ 20 less than its preceeding prize, find the value of each of the prizes.
- Ans. The first prize is ₹100, second prize is ₹ 80, third prize is ₹ 60 and fourth prize is ₹40.
Q. How many terms of the A.P. 9, 17, 25, … must be taken so that their sum is 636 ?
- Ans. 12.
- Ans. – 465.
- Ans. 5610.
Q. The first term of an A.P. is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.
- Ans. 16, 8/3
- Ans. 8n – 6, 34.
- Ans. 3, 1, 4.
Q. The first term of an A.P. is 17 and the last term is 350. If the common difference is 9, how many terms are there and what is their sum?
- Ans. 38, 6973.
- Ans. 6.