Advertisements
Advertisements
Question
The sum of first six terms of an arithmetic progression is 42. The ratio of the 10th term to the 30th term is `(1)/(3)`. Calculate the first and the thirteenth term.
Advertisements
Solution
T10 : T30 = 1 : 3, S6 = 42
Let a be the first term and d be a common difference, then
`(a + 9d)/(a + 29d) = (1)/(3)`
⇒ 3a + 27d = a + 29d
⇒ 3a – a = 29d – 27d
⇒ 2a = 2d
⇒ a = d
Now, S6 = 42
= `n/(2)[2a + (n - 1)d]`
⇒ 42 = `(6)/(2)[2a + (6 - 1)d]`
⇒ 42 = 3[2a + 5d]
⇒ 14 = 2a + 5d
⇒ 14 = 2a + 5a ...(∵ d = a)
⇒ 7a = 14
⇒ a = `(14)/(7)` = 2
∴ a = d = 2
Now, T13 = a + (n – 1)d
= 2 + (13 – 1) x 2
= 2 + 12 x 2
= 2 + 24
= 26
∴ 1st term is 2 and thirteenth term is 26.
APPEARS IN
RELATED QUESTIONS
Find the sum 2 + 4 + 6 ... + 200
Find the sum 3 + 11 + 19 + ... + 803
The sum of first n terms of an A.P is 5n2 + 3n. If its mth terms is 168, find the value of m. Also, find the 20th term of this A.P.
A man is employed to count Rs 10710. He counts at the rate of Rs 180 per minute for half an hour. After this he counts at the rate of Rs 3 less every minute than the preceding minute. Find the time taken by him to count the entire amount.
Mark the correct alternative in each of the following:
If 7th and 13th terms of an A.P. be 34 and 64 respectively, then its 18th term is
Q.12
A merchant borrows ₹ 1000 and agrees to repay its interest ₹ 140 with principal in 12 monthly instalments. Each instalment being less than the preceding one by ₹ 10. Find the amount of the first instalment.
The first term of an AP is –5 and the last term is 45. If the sum of the terms of the AP is 120, then find the number of terms and the common difference.
In the month of April to June 2022, the exports of passenger cars from India increased by 26% in the corresponding quarter of 2021-22, as per a report. A car manufacturing company planned to produce 1800 cars in 4th year and 2600 cars in 8th year. Assuming that the production increases uniformly by a fixed number every year. |
Based on the above information answer the following questions.
- Find the production in the 1st year
- Find the production in the 12th year.
- Find the total production in first 10 years.
[OR]
In how many years will the total production reach 31200 cars?
If the first term of an A.P. is p, second term is q and last term is r, then show that sum of all terms is `(q + r - 2p) xx ((p + r))/(2(q - p))`.
