Question 622864

Using the formula for an arithmetic sequence{{{t[n] = t[1] + (n - 1)d}}}, we get:


{{{t[10] = t[1] + (n - 1)d}}} 
{{{t[10] = t[1] + (10 - 1)d}}} ----- Substituting 10 for n in formula
{{{- 1 = t[1] + (10 - 1)d}}} 
{{{- 1 = t[1] + 9d}}}
{{{t[1] + 9d = - 1}}} ---- eq (i)


{{{t[n] = t[1] + (n - 1)d}}}
{{{t[20] = t[1] + (20 - 1)d}}} ----- Substituting 20 for n in formula
{{{7 = t[1] + (20 - 1)d}}} 
{{{7 = t[1] + 19d}}} 
{{{t[1] + 19d = 7}}} ------ eq (ii)


{{{t[1] + 9d = - 1}}} ---- eq (i)
{{{t[1] + 19d = 7}}} ---- eq (ii) 
- 10d = - 8 ------ Subtracting eq (ii) from eq (i)
d, or difference = {{{(- 8)/- 10}}}, or {{{4/5}}}


Since d = {{{4/5}}}, then {{{t[n] = t[1] + (n - 1)d}}} becomes: {{{t[10] = t[1] + (10 - 1) * (4/5)}}} ----- Substituting 10 for n, and {{{4/5}}} for d in formula
{{{- 1 = t[1] + 9 * (4/5)}}}
{{{- 1 = t[1] + (36/5)}}}
{{{t[1] = - 1 - (36/5 )}}}
{{{t[1] = - (5/5) - (36/5)}}}
Initial term, or {{{highlight_green(t[1] = - (41/5))}}}


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Check
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10th term
{{{t[n] = t[1] + (n - 1)d}}} 
{{{- 1 = (- 41/5) + (10 - 1) * (4/5)}}}
{{{- 1 = (- 41/5) + 9 * (4/5)}}}
{{{- 1 = (- 41/5) + (36/5)}}}
{{{- 1 = - (5/5)}}}
- 1 = - 1 (TRUE)


20th term
{{{t[n] = t[1] + (n - 1)d}}} 
{{{7 = (- 41/5) + (20 - 1) * (4/5)}}}
{{{7 = (- 41/5) + 19 * (4/5)}}}
{{{7 = (- 41/5) + (76/5)}}}
{{{7 = (35/5)}}}
7 = 7 (TRUE)


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