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 of the following APs:
0.6, 1.7, 2.8, ... to 100 terms.
In a school, students thought of planting trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant will be the same as the class, in which they are studying, e.g., a section of class I will plant 1 tree, a section of class II will plant 2 trees, and so on till class XII. There are three sections of each class. How many trees will be planted by the students?
If the sum of first m terms of an A.P. is the same as the sum of its first n terms, show that the sum of its first (m + n) terms is zero
The first and the last terms of an A.P. are 34 and 700 respectively. If the common difference is 18, how many terms are there and what is their sum?
The fourth term of an A.P. is 11 and the eighth term exceeds twice the fourth term by 5. Find the A.P. and the sum of first 50 terms.
Find the sum of first n terms of an AP whose nth term is (5 – 6n). Hence, find the sum of its first 20 terms.
If Sn denotes the sum of first n terms of an A.P., prove that S12 = 3(S8 – S4).
If Sn denote the sum of the first n terms of an A.P. If S2n = 3Sn, then S3n : Sn is equal to
The sum of all two digit odd numbers is ______.
Kanika was given her pocket money on Jan 1st, 2008. She puts Rs 1 on Day 1, Rs 2 on Day 2, Rs 3 on Day 3, and continued doing so till the end of the month, from this money into her piggy bank. She also spent Rs 204 of her pocket money, and found that at the end of the month she still had Rs 100 with her. How much was her pocket money for the month?
