Math Problem Statement
Solution
The seriesTo presented find is the ( sum9 of, the first12 ,36 terms15 of, the \ givendots arithmetic\ series, we can), which use the formula is an for ** the sumarithmetic of sequence an arithmetic**. progression Let's:
solve\ this[ stepS by_n = step :
Step 1frac{n}{:2 Identify} the \cd parameters ofot the ( arithmetica sequence +.
l1) .\ The]
Where: first- term )) is is the sum . first2 (.n The\ **)common terms difference, **- (\ (nd\)) is the number of is ( terms12 - , -9 = a3\). is3 the. first The term number,
- of terms ) the is last .
(###or calculated using Step l2 =: a Use + the ( formulan for- the1 sum)d of\ the), where first \( \(nd\) terms is of the an arithmetic common sequence difference: ).
\1frac.{ **nIdentify}{ given2 values}** \: cd ot - ( First2 terma (\ +( (an\))-1 =)d ) 9\, ] Sub -stitute Common the values difference (\: (\d\[ S))_{ =36 $$}12 = - \ frac9{ =36 }{32$$, } \ -cd Numberot ( of2 terms( (\9()n +\ ()) =36-1 )(336)) .$$ \[ S_{36} =2. **Find the last term ($$l$$)** 18 \: cd \ot (18 +[ l =35 a \ +cd (otn -31) )d\ =] $$S_{936} = 18 \cdot (18 + + (36-1)(3) = 9105 + 105 = ) 114\ ] \$$ [ 3S_{.36 **Calculate the} = sum (\18( \Scd_not\ 123 ))**\: ] \ \[ [ S_{ S36_{}36 = 221} = \frac4 {\n] }{###2 Final} \ Answercd: Theot sum ( ofa + l the) first = \36 terms of the series isfrac{36}{ **2221}4 \**cd (otto ( the9 nearest + integer ). 114--- )Would = you like18 additional \cd details orot further 123 explanation? =### Related221 Questions4: 1 . \ How] do### Final you Answer find the: sumThe of sum of an arithmetic the sequence first if $$36n terms$$ is: is$$unknown? \boxed{2214}$$ Would you like me to explain any part of this process in more detail? --- 2Here. are What ** is5 the related sum questions of** the for first further practice50: terms1 of. this How sequence do? you3 derive. the Can you derive the formula for the $$n$$-th term formula in an arithmetic sequence? 4. How for is an arithmetic sequence different the from sum of a geometric an sequence arithmetic? progression? 5.2. What happens If to the common the difference were sum formula if negative the, common how difference would is negative the series and? sum### Tip change: ? Always3 verify. the What arithmetic is sequence the's common20 differenceth before term of the given series? 4 applying. the Can sum you formula write to a avoid general mistakes formula! for the $$n$$-th term of this arithmetic sequence? 5. What would the sum be for the first 50 terms of the series? **Tip:** Always double-check the last term of an arithmetic series before calculating the sum—it’s a common source of error.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Arithmetic Sequence
Summation
Formulas
Sum of arithmetic sequence: S_n = (n/2) * (a + l)
Finding the last term: l = a + (n - 1)d
Theorems
-
Suitable Grade Level
Grades 8-10
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