Find the sum of the following series.
(i) $1 + 2 + 3 + . . . . + 45$
(ii) $16^{2} + 17^{2} + 18^{2} + . . . . + 25^{2}$
(iii) $2 + 4 + 6 + . . . . + 100$
(iv) $7 + 14 + 21 + . . . . + 490$
(v) $5^{2} + 7^{2} + 9^{2} + 39^{2}$
(vi) $16^{3} + 17^{3} + . . . . . + 35^{3}$

Open in App

Solution

(i) $1 + 2 + 3.. + 45$ $= \frac{45 (45 + 1)}{2} = 1035$
(ii) $16^{2} + 17^{2} . . + 25^{2}$ $= 1^{2} + 2^{2} + 3^{2} . . . 25^{2} - (1^{2} + 2^{2} . . . 15^{2}) = \frac{25 (26) (51)}{6} - \frac{15 (16) (31)}{6} = 5525 - 1240 = 4285$
(iii) $2 + 4 + 6.. + 100$ $= 2 (1 + 2 + 3... + 50) = 2 \times \frac{50 (51)}{2} = 2550$
(iv) $7 + 14 + . . . + 490$ $= 7 (1 + 2 + 3... + 70) = 7 \times \frac{70 (71)}{2} = 17395$
(v) $5^{2} + 7^{2} + 9^{2} + 39^{2} = 1676$
(vi) $16^{3} + 17^{3} . . . + 35^{3}$ $= 1^{3} + 2^{3} + . . . 35^{3} - (1^{3} + 2^{3} + 3^{3} . . . + 15^{3}) = {(\frac{35 (36)}{2})}^{2} - {(\frac{15 (16)}{2})}^{2} = 396900 - 14400 = 382500$

Suggest Corrections

Similar questions

Match Column A appropriately with Column B.

$\begin{matrix} C o l u m n A & C o l u m n B (i)The Central Nervous System constitutes of & 1.Ears,Eyes,Nose (ii)Reflex axes are formed in & 2.Olfactory (iii)Receptors that detect taste & 3. Spinal cord and the Brain (iv)Receptors that detect smell & 4. Muscles and Glands (v)Examples of receptor organs are & 5. Gustatory (vi)Examples of effector organs are & 6. Spinal Cord (vii)Peripheral Nervous System consists of & 7. Spinal nerves from the spinal cord and cranial nerves from the brain \end{matrix}$

Match the following

(1)–(iii), (viii); (2)–(i), (iv); (3)–(ii), (vi); (4)-(ii),(v); (5)–(iii), (iv); (6)–(i) (vii); (7)–(ii), (iv)
(1)–(ii), (vii); (2)–(i), (iv); (3)–(ii), (vi); (4)-(ii),(v); (5)–(ii), (iv); (6)–(i) (viii); (7)–(iii), (iv)
(1)–(ii), (vii); (2)–(i), (vii); (3)–(iii), (iv); (4)-(i),(v); (5)–(ii), (vi); (6)–(i) (vi); (7)–(i), (iv)
(1)–(ii), (vii); (2)–(i), (iv); (3)–(ii), (vi); (4)-(ii),(iv); (5)–(i), (viii); (6)–(iii) (iv); (7)–(iii), (vii)

Q. Find the values of n in each of the following:
(i)

5^{2 n} \times 5^{3} = 5^{11}

(ii)

9 \times 3^{n} = 3^{7}

(iii)

8 \times 2^{n + 2} = 32

(iv)

7^{2 n + 1} \div 49 = 7^{3}

(v)

{(\frac{3}{2})}^{4} \times {(\frac{3}{2})}^{5} = {(\frac{3}{2})}^{2 n + 1}

(vi)

{(\frac{2}{3})}^{10} + {\{{(\frac{3}{2})}^{2}\}}^{5} = {(\frac{2}{5})}^{2 n - 2}

Q. Find the value of:
(i)

(3^{0} + 4^{- 1}) \times 2^{2}

(ii)

(2^{- 1} \times 4^{- 1}) \div 2^{- 2}

(iii)

{(\frac{1}{2})}^{- 2} + {(\frac{1}{3})}^{- 2} + {(\frac{1}{4})}^{- 2}

(iv)

(3^{- 1} + 4^{- 1} + 5^{- 1})^{0}

(v)

{[{(\frac{- 2}{3})}^{- 2}]}^{2}

(vi)

7^{- 20} - 7^{- 21}

Match Column A appropriately with Column B.

$\begin{matrix} C o l u m n A & C o l u m n B (i)The Central Nervous System constitutes of & 1.Ears,Eyes,Nose (ii)Reflex arcs are formed in & 2.Olfactory (iii)Receptors that detect taste & 3. Spinal cord and the Brain (iv)Receptors that detect smell & 4. Muscles and Glands (v)Examples of receptor organs are & 5. Gustatory (vi)Examples of effector organs are & 6. Spinal Cord (vii)Peripheral Nervous System consists of & 7. Spinal nerves from the spinal cord and cranial nerves from the brain \end{matrix}$

Join BYJU'S Learning Program

Find the sum of the following series.(i) 1+2+3+....+45(ii) 162+172+182+....+252(iii) 2+4+6+....+100(iv) 7+14+21+....+490(v) 52+72+92+392(vi) 163+173+.....+353

Find the sum of the following series.
(i) $1 + 2 + 3 + . . . . + 45$
(ii) $16^{2} + 17^{2} + 18^{2} + . . . . + 25^{2}$
(iii) $2 + 4 + 6 + . . . . + 100$
(iv) $7 + 14 + 21 + . . . . + 490$
(v) $5^{2} + 7^{2} + 9^{2} + 39^{2}$
(vi) $16^{3} + 17^{3} + . . . . . + 35^{3}$