Sum of First 20 Terms of Arithmetic Series: 131+235

[UnD]

Lol it's quite easy but these questions r annoying me.
The 20th term of an arithmetic series if 131 and sum of the 6th to 10th term inclusive is 235, find sum is the first 20 terms
well
131= a + 19 d 5(2a+ 9d) -3(2a+5d)=235
well i just went on and solve simutalneosly and got the wrong answer for S(20), i don't know how to make it at hte butotm.
2) the sum of 50 terms of an arithmetic is 249 and sum of 49 terms is 233, find 50th term of the series.
249= 50a + 1225d and 233= 49/2 (2a+48d)
and well got the wrong answer aagain.
3) prove $T_n = S_n - S_n_-_1$
I have no idea how to do that. Thanks

 

[VietDao29]

...
well
131= a + 19 d
5(2a+ 9d) -3(2a+5d)=235
well i just went on and solve simutalneosly and got the wrong answer for S(20), ...

I have bolded the wrong equation in your post. In fact the sum of the sixth term to the tenth term is:
S_{6 - 10} = S₁₀ - S₅ = $$5(2a + 9d) - \frac{5}{2}(2a + 4d) = 235$$
not
S_{6 - 10} = S₁₀ - S₆

2) the sum of 50 terms of an arithmetic is 249 and sum of 49 terms is 233, find 50th term of the series.
249= 50a + 1225d
233= 49/2 (2a+48d)

You are complicating the problem...
S₅₀ = a₁ + a₂ + a₃ + ... + a₄₉ + a₅₀
S₄₉ = a₁ + a₂ + a₃ + ... + a₄₉
So what's a₅₀?

3) prove $T_n = S_n - S_n_-_1$
I have no idea how to do that. Thanks

I have no idea what T_n is... Is that the n-th term of the series?
Viet Dao,

 

[borisleprof]

your second equation is wrong. it should read 5a + 35d = 235; after that you can solve the system formed withthis equation et your first equation to find a and d
After that it should be easy to find the sum of the first 20 terms.

 

[ivybond]

The 20th term of an arithmetic series if 131 and sum of the 6th to 10th term inclusive is 235, find sum is the first 20 terms

Concept of mean is often helpful in questions on arithmetic sequences.
Mean of n first terms of such sequence is
a_mean = S_n/n
If n is odd, a_mean is one of the terms of the sequence.
Which one?
Knowing that
a₆ + a₇ + a₈ + a₉ + a₁₀ = 235
you can determine the value of a_?.
Now you can find a₁ from a_? and a₂₀ in one step.
a₁ and a₂₀ lead you straight to S₂₀.