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Standard XII

Mathematics

Sum of n Terms

If the sum of...

Question

If the sum of the first $n$ terms of an AP is $4 n - n^{2}$ , what is the first term (that is $S_{1}$ ) What is the sum of first two terms? What is the second term? Similarly, find the $3 r d$ , the $10 t h$ and the nth terms.

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Solution

$s_{n} = 4 n - n^{2}$
$T_{n} = s_{n} - s_{n - 1}$
$T_{n} = 4 n - n^{2} - 4 (n - 1) + (n - 1)^{2}$
$T_{n} = 4 n - n^{2} - 4 n + 4 + n^{2} - 2 n + 1$
$T_{n} = - 2 n + 5$
$T_{1} = - 2 \times 1 + 5 = 3$
$T_{2} = - 2 \times 2 + 5 = 1$
$T_{3} = - 2 \times 3 + 5 = - 1$
$T_{10} = - 2 \times 10 + 5 = - 15$
sum of first two terms $= T_{1} + T_{2} = 3 + 1 = 4$
second term $= T_{2} = 1$
third term $= T_{3} = - 1$
tenth term $= t_{10} = - 15$

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If the sum of the first n terms of an AP is 4n−n2, what is the first term (that is S1) What is the sum of first two terms? What is the second term? Similarly, find the 3rd, the 10th and the nth terms.

sn=4n−n2Tn=sn−sn−1Tn=4n−n2−4(n−1)+(n−1)2Tn=4n−n2−4n+4+n2−2n+1Tn=−2n+5T1=−2×1+5=3T2=−2×2+5=1T3=−2×3+5=−1T10=−2×10+5=−15sum of first two terms=T1+T2=3+1=4second term=T2=1third term=T3=−1tenth term=t10=−15

If the sum of the first $n$ terms of an AP is $4 n - n^{2}$ , what is the first term (that is $S_{1}$ ) What is the sum of first two terms? What is the second term? Similarly, find the $3 r d$ , the $10 t h$ and the nth terms.