+0

arithmetric sequence

0
448
2

a22 = -49 and a25 = -58

Find a1

Guest Jun 17, 2015

#2
+19207
+10

$$\small{\text{ \mathbf{ a_{22} = -49 \quad \mathrm{and} \quad a_{25}= -58 \quad \mathrm{find} \quad a_1 } }}$$

$$\mathbf{ \mathrm{Explicit~ Formula:} & \quad a_n = a_1 + (n-1)\cdot d } \\\\ \small{\text{ \begin{array}{lrcl} & a_{22} &=& a_1 + 21 \cdot d \\ & a_{25} &=& a_1 + 24 \cdot d \\ \\ \hline \\ & a_{22} - a_1 &=& 21 \cdot d \\ & a_{25} - a_1 &=& 24 \cdot d \\ \\ \hline \\ & \dfrac{ a_{22} - a_1 } { a_{25} - a_1 } &=& \dfrac{21} {24}\\ \\ & \dfrac{ a_{22} - a_1 } { a_{25} - a_1 } &=& \dfrac{7} {8}\\\\ & 8\cdot ( a_{22} - a_1 ) &=& 7\cdot ( a_{25} - a_1 )\\ & 8\cdot a_{22} - 8\cdot a_1 &=& 7\cdot a_{25} - 7\cdot a_1\\ & 8\cdot a_1 - 7\cdot a_1 &=& 8\cdot a_{22} - 7\cdot a_{25} \\ & \mathbf{a_1} &\mathbf{=}& \mathbf{8\cdot a_{22} - 7\cdot a_{25}} \\\\ & a_1 &=& 8\cdot (-49) - 7 \cdot (-58 ) \\ & a_1 &=& -392 + 406 \\ & \mathbf{a_1} &\mathbf{=}& \mathbf{14} \end{array} }}$$

heureka  Jun 18, 2015
Sort:

#1
+85759
+5

-58 - (-49)  = -9     is the difference in these terms which seems to imply that  -9/3 = -3 is the common difference between terms

So......

-58  = a1 + d(n - 1)

-58   = a1 - 3(25 - 1)

-58  = a1 - 72    add 72 to both sides

14  = a1

CPhill  Jun 18, 2015
#2
+19207
+10

$$\small{\text{ \mathbf{ a_{22} = -49 \quad \mathrm{and} \quad a_{25}= -58 \quad \mathrm{find} \quad a_1 } }}$$

$$\mathbf{ \mathrm{Explicit~ Formula:} & \quad a_n = a_1 + (n-1)\cdot d } \\\\ \small{\text{ \begin{array}{lrcl} & a_{22} &=& a_1 + 21 \cdot d \\ & a_{25} &=& a_1 + 24 \cdot d \\ \\ \hline \\ & a_{22} - a_1 &=& 21 \cdot d \\ & a_{25} - a_1 &=& 24 \cdot d \\ \\ \hline \\ & \dfrac{ a_{22} - a_1 } { a_{25} - a_1 } &=& \dfrac{21} {24}\\ \\ & \dfrac{ a_{22} - a_1 } { a_{25} - a_1 } &=& \dfrac{7} {8}\\\\ & 8\cdot ( a_{22} - a_1 ) &=& 7\cdot ( a_{25} - a_1 )\\ & 8\cdot a_{22} - 8\cdot a_1 &=& 7\cdot a_{25} - 7\cdot a_1\\ & 8\cdot a_1 - 7\cdot a_1 &=& 8\cdot a_{22} - 7\cdot a_{25} \\ & \mathbf{a_1} &\mathbf{=}& \mathbf{8\cdot a_{22} - 7\cdot a_{25}} \\\\ & a_1 &=& 8\cdot (-49) - 7 \cdot (-58 ) \\ & a_1 &=& -392 + 406 \\ & \mathbf{a_1} &\mathbf{=}& \mathbf{14} \end{array} }}$$

heureka  Jun 18, 2015

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