Series and Sequences - Sum of a series (1 Viewer)

[Lucas_]

Hey, can anybody please help me with this moderately difficult question? If you could do a simple solution with explanations I would be pretty grateful.

The 6th term of an arithmetic series is 23 and the sum of the first 10 terms is 210. Find the sum of 20 terms.

Thanks

 

[qwerty44]

Use simultaneous equations with variable a and d.

Then when you solve for a and d just find the sum when n=20

i.e.
a+5d=23
a=23-5d (1)

10/2(2a+9d)=210 (2)


sub (1) in (2)

Solve.

 

[MetalTheory]

First, you simplify the equations including the parameters which you need to find:

T₆ = 23 = a + (6-1).d

= a + 5d -(1)​

S₁₀ = 210 = 10/2 [2a + (10-1).d]

= 5(2a + 9d)​

42 = 2a + 9d -(2)​

(1)*2

46 = 2a + 10d -(3)

Then, using simultaneous equations, find a and d.

(3)-(2)

d=4

substitute d into (1)

23 = a + 5(4)

= a + 20​

a = 3

Finally, you substitute the values just found into the arithmetic sum equation.

S₂₀ = 20/2 [2(3) + (20-1).4]

= 10(82)​

= 820​

Hopefully that makes sense.

EDIT: Well, looks like someone beat me to it.

 

[xDarkSilent]

Both correct
Lucas, Just remember your equations
Tn , Sn , S(infinity) . And youll be set for any of these questions

Best of luck !

 

[Lucas_]

First, you simplify the equations including the parameters which you need to find:

T₆ = 23 = a + (6-1).d

= a + 5d -(1)​

S₁₀ = 210 = 10/2 [2a + (10-1).d]

= 5(2a + 9d)​

42 = 2a + 9d -(2)​

(1)*2

46 = 2a + 10d -(3)

Then, using simultaneous equations, find a and d.

(3)-(2)

d=4

substitute d into (1)

23 = a + 5(4)

= a + 20​

a = 3

Finally, you substitute the values just found into the arithmetic sum equation.

S₂₀ = 20/2 [2(3) + (20-1).4]

= 10(82)​

= 820​

Hopefully that makes sense.

EDIT: Well, looks like someone beat me to it.

Hey, I'm not understanding why you have multiplied (1) x 2 in this solution before you solved the simultaneous equations?

Thanks for all the responses!

 

[Carrotsticks]

So then the Elimination Method can be used between (1) and (2)

 

[Lucas_]

Yeah, I think that makes more sense now carrotsticks.

The worked solution I was looking through had an error; It had (1) multiplied by (2) not by 2.

Thanks all.