Advertisements
Advertisements
प्रश्न
In an A.P., if the 5th and 12th terms are 30 and 65 respectively, what is the sum of first 20 terms?
Advertisements
उत्तर
In the given problem, let us take the first term as a and the common difference d
Here, we are given that,
`a_5 = 30` ....(1)
`a_12 = 65` .....(2)
Also, we know
`a_n = a + (n - 1)d`
For the 5th term (n = 5)
`a_5 = a + (5 - 1)d`
30 = a + 4d (Using 1)
a = 30 - 4d ....(3)
Similarly for the 12 th term (n = 12)
`a_12 = a + (12 - 1)d`
65 = a + 11d (Using 2)
a = 65 - 11d....(4)
Substracting (3) from (4) we get
a - a = (65 - 11d)-(30 - 4d)
0 = 65 - 11d - 30 + 4d
0 = 35 - 7d
7d = 35
d = 5
Now, to find a, we substitute the value of d in (4),
a = 30 - 4(5)
a = 30 - 20
a = 10
So for the given A.P d = 5 and a = 10
So to find the sum of first 20 terms of this A.P. we use the following formula for the sum of n terms of an A.P
`S_n = n/2 [2a + (n - 1)d]`
Where a = first term for the given A.P
d= common difference of the given A.P
n = number of terms
So using the formula for n = 20 we get
`S_20 = 20/2 [2(10) + (20 - 1)(5)]`
= (10)[20 + (19)(5)]
= (10)[20 + 95]
= (10)[115]
= 1150
Therefore, the sum of first 20 terms for the given A.P. is `S_20 = 1150`
APPEARS IN
संबंधित प्रश्न
The first and the last terms of an AP are 8 and 65 respectively. If the sum of all its terms is 730, find its common difference.
The sum of the third and the seventh terms of an AP is 6 and their product is 8. Find the sum of first sixteen terms of the AP.
A small terrace at a football field comprises 15 steps, each of which is 50 m long and built of solid concrete. Each step has a rise of `1/4` m and a tread of `1/2` m (See figure). Calculate the total volume of concrete required to build the terrace.
[Hint: Volume of concrete required to build the first step = `1/4 xx 1/2 xx 50 m^3`]
Find the sum of n terms of an A.P. whose nth terms is given by an = 5 − 6n.
Find the sum of all odd natural numbers less than 50.
If 4 times the 4th term of an A.P. is equal to 18 times its 18th term, then find its 22nd term.
The sum of the first n terms in an AP is `( (3"n"^2)/2 +(5"n")/2)`. Find the nth term and the 25th term.
How many terms of the A.P. 21, 18, 15, … must be added to get the sum 0?
Write an A.P. whose first term is a and common difference is d in the following.
a = –19, d = –4
The sum of 5th and 9th terms of an A.P. is 30. If its 25th term is three times its 8th term, find the A.P.
Write 5th term from the end of the A.P. 3, 5, 7, 9, ..., 201.
Write the expression of the common difference of an A.P. whose first term is a and nth term is b.
The common difference of the A.P. is \[\frac{1}{2q}, \frac{1 - 2q}{2q}, \frac{1 - 4q}{2q}, . . .\] is
Q.13
Q.19
Find the sum of first 10 terms of the A.P.
4 + 6 + 8 + .............
In an A.P. (with usual notations) : given d = 5, S9 = 75, find a and a9
The sum of first n terms of an A.P. whose first term is 8 and the common difference is 20 equal to the sum of first 2n terms of another A.P. whose first term is – 30 and the common difference is 8. Find n.
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.
Which term of the Arithmetic Progression (A.P.) 15, 30, 45, 60...... is 300?
Hence find the sum of all the terms of the Arithmetic Progression (A.P.)
