Arithmetic Sequences - T5 = 3T2 and S6= 36, find the first three terms.

**Dev** · October 26th 2008, 05:37 AM

Yo, some questions.

As in how do u do these. Thanks.

T5 = 3T2 and S6= 36, find the first three terms.

also

Sn = 5n2(sqaured that is) + 3n - Find the first three terms and thus find a formula for Tn

Sweet as Thanks.

**james_bond** · October 26th 2008, 06:19 AM

$a_5=a_2+(5-2)d$ and we know that $a_5=3a_2$ so $2a_2=3d$ . And we were also given: $36=S_6=\frac {6(a_1+a_6)}2=3((a_2-d)+(a_2+4d))=3(2a_2+3d)=18d\rightarrow d=2$ . Using this: $a_2=6$ . So the first three terms: $4, 6, 8$ .

For the second problem: $S_n=n(3+5n)=\frac{n(6+10n)}2=\frac{n(6+10(n-1)+10)}2=\frac{n(2\cdot 8+(n-1)\cdot 10)}2$ . You can see that $a_1=8$ and $d=10$ therefore $a_n=8+(n-1)\cdot 10$ and the first three terms: $8, 18, 28$ .

**Soroban** · October 26th 2008, 06:43 AM

Hello, Dev!

We are expected to know these formulas:

. . $n^{th}\text{ term: }\;T_n \:=\:a + (n-1)d$

. . $\text{Sum of first }n\text{ terms: }\;S_n \;=\;\frac{n}{2}[2a + (n-1)d]$

. . . . where: . $\begin{Bmatrix}a &=& \text{first term} \\ d &=& \text{common diff.} \\ n &=& \text{no. of terms} \end{Bmatrix}$

$T_5 \:=\: 3\!\cdot\!T_2\:\text{ and }\:S_6\:=\: 36$
Find the first three terms.

2nd term: . $T_2 \:=\:a+d$
5th term: . $T_5 \:=\:a+4d$

$T_5 = 3\!\cdot\!T_2 \quad\Rightarrow\quad a + 4d \:=\:3(a+d) \quad\Rightarrow\quad d \:=\:2a$ .[1]

$S_6 = 36\!:\;\;36 \:=\:\frac{6}{2}[2a + 5d] \:=\:36 \quad\Rightarrow\quad 2x + 5d \:=\:12$ .[2]

Substitute [1] into [2]: . $2a + 5(2a) \:=\:12 \quad\Rightarrow\quad 12a \:=\:12 \quad\Rightarrow\quad\boxed{ a \:=\:1}$

Substitute into [1]: . $d = 2(1) \quad\Rightarrow\quad\boxed{d \:= \:2}$

The first term is 1, the common difference is 2.

Therefore, the first three terms are: .1, 3, 5.

**Soroban** · October 26th 2008, 07:09 AM

Hello again, Dev!

$S_n \:=\: 5n^2 + 3n$
Find the first three terms and thus find a formula for $T_n$

The sum of the first $n$ terms is: . $S_n \:=\:\frac{n}{2}[2a + (n-1)d]$

The sum of the first one term is: . $a$
The formula says: . $S_1 \:=\:5(1^2) + 3(1) \:=\:8$
. . Hence: . $\boxed{ a\:=\:8}$ .[1]

The sum of the first two terms is: . $a + (a+d) \:=\:2a+d$
The formula says: . $S_2 \:=\:5(2^2) + 3(2) \:=\:26$
. . Hence: . $2a+d \:=\:26$ .[2]

Substitute [1] into [2]: . $2(8) + d \:=\:26 \quad\Rightarrow\quad\boxed{ d \:=\:10}$

The first term is: $a = 8$ . . . the common difference is: $d = 10$

$\text{The first three terms are: }\;{\bf{\color{red}8,\:18,\:28}}$

$\text{The }n^{th}\text{ term is: }\;T_n \:=\:8 +10(n-1) \quad\Rightarrow\quad {\bf{\color{red} T_n \:=\:10n -2}}$

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