Suppose a1, a2, … an, … be an arithmetic progression of natural numbers. If the ration of the sum of first five terms to the sum of first nine terms of the progression is 5 : 17 and 110 < a15 < 120, then the sum of the first ten terms of the progression is equal to
- 290
- 380
- 460
- 510
The Correct Option is B
Solution and Explanation
The correct answer is (B):
∵ a1, a2, … an … be an A.P of natural numbers and
\(\frac{S_5}{S_9} =\frac{ 5}{17}\)
⇒ \(\frac{\frac{5}{2}[2α_1+4d]}{\frac{9}{2}[2α_1+8d]} = \frac{5}{17}\)
⇒ 34α1+68d
= 18α1+72d
∴ d = 4α1
And 110<α15<120
∴ 110<α1+14d<120
⇒ 110<57α1<120
∴ α1 = 2(∵ αi ∈N)
d = 8
∴ S10 = 5[4+9×8]
= 380
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Concepts Used:
Types of Differential Equations
There are various types of Differential Equation, such as:
Ordinary Differential Equations:
Ordinary Differential Equations is an equation that indicates the relation of having one independent variable x, and one dependent variable y, along with some of its other derivatives.
\(F(\frac{dy}{dt},y,t) = 0\)
Partial Differential Equations:
A partial differential equation is a type, in which the equation carries many unknown variables with their partial derivatives.
Linear Differential Equations:
It is the linear polynomial equation in which derivatives of different variables exist. Linear Partial Differential Equation derivatives are partial and function is dependent on the variable.
Homogeneous Differential Equations:
When the degree of f(x,y) and g(x,y) is the same, it is known to be a homogeneous differential equation.
\(\frac{dy}{dx} = \frac{a_1x + b_1y + c_1}{a_2x + b_2y + c_2}\)
Read More: Differential Equations