How many terms with should be added in the series 1+5+9+13.....to get 190
Answers
Answer:
10
Step-by-step explanation:
Given terms are in AP. This means they have a common difference.
Sum of first n terms of AP = (n/2)[ 2a + ( n - 1 )d ], where a is the first term and d is the common difference.
Here,
a = first term = 1
d = co. di. = 5 - 1 = 4
Let 190 be the nth term so
⇒ 190 = (n/2)[ 2(1) + ( n - 1 )4 ]
⇒ 380 = n( 2 - 4 + 4n )
⇒ 380 = ( 4n - 2 )n
⇒ 190 = 2n^2 - n
⇒ 2n^2 - n - 190 = 0
⇒ 2n^2 - 20n + 19n - 190 = 0
= > ( n - 10 )( 2n + 19 ) = 0
= > n = 10
Series : 1 + 5 + 9 + 13 + ......
a = 1
d = 5 - 1 = 4
Sₙ = 190
n = ?
Sₙ = n/2 [ 2a + (n - 1) d ]
190 = n/2 [ 2 + (n - 1) 4 ]
380 = n [ 2 + 4n - 4 ]
380 = n [ 4n - 2 ]
4n² - 2n - 380 = 0
2n² - n - 190 = 0
2n² - 20n + 19n - 190 = 0
2n( n - 10 ) + 19 ( n - 10 ) = 0
( n - 10 ) ( 2n + 19 ) = 0
n - 10 = 0 or 2n + 19 = 0
n= 10 , n = -19/2
But n can not be in fraction nor negative.
n = 10
Hence, 10 terms are required for a sum of 190.