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Bess [88]
3 years ago
6

Show work and explain with formulas.

Mathematics
1 answer:
vova2212 [387]3 years ago
4 0

Answer:

\large\boxed{4.\ S_{13}=260}

\large\boxed{5.\ \sum\limits_{k=1}^7(2k+5)=91}

\large\boxed{6.\ a_1=17,\ a_2=26,\ a_3=35}

Step-by-step explanation:

4.

We have:

a_6=18,\ a_{13}=32

These are the terms of the arithmetic sequence.

We know:

a_n=a_1+(n-1)d

Therefore

a_6=a_1+(6-1)d\ \text{and}\ a_{13}=a_1+(13-1)d\\\\a_6=a_1+5d\ \text{and}\ a_{13}=a_1+12d\\\\a_{13}-a_6=(a_1+12d)-(a_1+5d)\\\\a_{13}-a_6=a_1+12d-a_1-5d\\\\a_{13}-a_6=7d

Substitute a₆ = 18 and a₁₃ = 32:

7d=32-18

7d=14             <em>divide both sides by 7</em>

d=2

a_6=a_1+5d

18=a_1+5(2)

18=a_1+10            <em>subtract 10 from both sides</em>

8=a_1\to a_1=8

The formula of a sum of terms of an arithmetic sequence:

S_n=\dfrac{a_1+n_1}{2}\cdot n

Substitute a₁ = 8, a₁₃ = 32 and n = 13:

S_{13}=\dfrac{8+32}{2}\cdot13=\dfrac{40}{2}\cdot13=20\cdot13=260

===========================================

5.

We have

\sum\limits_{k=3}^7(2k+5)\to a_k=2k+5

Calculate a_{k+1}

a_{k+1}=2(k+1)+5=2k+2+5=2k+7

Calculate the difference:

a_{k+1}-a_k=(2k+7)-(2k+5)=2k+7-2k-5=2

It's the arithmetic sequence with first term

a_1=2(1)+5=2+5=7

and common difference d = 2.

The formula of a sum of terms of an arithmetic sequence:

S_n=\dfrac{2a_1+(n-1)d}{2}\cdot n

Substitute n = 7, a₁ = 7 and d = 2:

S_7=\dfrac{2(7)+(7-1)(2)}{2}\cdot7=\dfrac{14+(6)(2)}{2}\cdot7=\dfrac{14+12}{2}\cdot7=\dfrac{26}{2}\cdot7=(13)(7)\\\\S_7=91

===========================================

6.

We have:

a_1=17,\ a_n=197,\ S_n=2247

The formula for the n-th term of an arithmetic sequence:

a_n=a_1+(n-1)d

The formula of the sum of terms of an arithmetic sequence:

S_n=\dfrac{2a_1+(n-1)d}{2}\cdot n

Substitute:

(1)\qquad 197=17+(n-1)d\\\\(2)\qquad2247=\dfrac{(2)(17)+(n-1)d}{2}\cdot n

Convert the first equation:

197=17+(n-1)d          <em>subtract 17 from both sides</em>

180=(n-1)d    Substitute it to the second equation:

2247=\dfrac{34+180}{2}\cdot n\\\\2247=\dfrac{214}{2}\cdot n

2247=107n             <em>divde both sides by 107</em>

21=n\to n=21

Put the value of n to the equation (n - 1)d = 180:

(21-1)d=180

20d=180              <em>divide both sides by 20</em>

d=9

Therefore we have the explicit formula for the nth term of an arithmetic sequence:

a_n=17+(n-1)(9)\\\\a_n=17+9n-9\\\\a_n=9n+8

Put n = 1, n = 2 and n = 3:

a_1=9(1)+8=9+8=17\qquad\text{CORRECT :)}\\\\a_2=9(2)+8=18+8=26\\\\a_3=9(3)+8=27+8=35

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