The sum 1 ^ 2 - 2 . 3 ^ 2 + 3 . 5 ^ 2 - 4 . 7 ^ 2 + 5 . 9 ^ 2 - . ....

Given,

$S = {1}^{2} - 2 \cdot {3}^{2} + 3 \cdot {5}^{2} . . . . . . . . . . . . + 15 \cdot {29}^{2}$

Now rewriting the above expression we get,

$S = \mathop{\mathop{{1}^{2} + 2 \cdot {3}^{2} + 3 \cdot {5}^{2} . . . . . . . . . . . . + 15 \cdot {29}^{2}}_{⏟}}_{{S}_{1}} - 2\mathop{\mathop{\left[2 \cdot {3}^{2} + 4 \cdot {7}^{2} . . . . . . . . . . . . + 14 \cdot {27}^{2}\right]}_{⏟}}_{{S}_{2}}$

Now solving ${S}_{1} = {1}^{2} + 2 \cdot {3}^{2} + 3 \cdot {5}^{2} . . . . . . . . . . . . + 15 \cdot {29}^{2}$

$⇒ {S}_{1} = \mathop{∑}_{r = 1}^{15}r \cdot {\left(2r - 1\right)}^{2}$

$⇒ {S}_{1} = \mathop{∑}_{r = 1}^{15}r \cdot \left(4{r}^{2} + 1 - 4r\right)$

$⇒ {S}_{1} = \mathop{∑}_{r = 1}^{15}\left(4{r}^{3} + r - 4{r}^{2}\right)$

$⇒ {S}_{1} = \left(4\mathop{∑}_{r = 1}^{15}{r}^{3} + \mathop{∑}_{r = 1}^{15}r - 4\mathop{∑}_{r = 1}^{15}{r}^{2}\right)$
$⇒ {S}_{1} = \left(4{\left(\dfrac{15 \times 16}{2}\right)}^{2} + \dfrac{15 \times 16}{2} - 4\dfrac{15 \times 16 \times 31}{6}\right)$

$⇒ {S}_{1} = \left(4{\left(120\right)}^{2} + 120 - 4960\right)$

$⇒ {S}_{1} = 52760$

Now solving ${S}_{2} = 2 \cdot {3}^{2} + 4 \cdot {7}^{2} . . . . . . . . . . . . + 14 \cdot {27}^{2}$

$⇒ {S}_{2} = \mathop{∑}_{r = 1}^{7}\left(2r\right){\left(4r - 1\right)}^{2}$

$⇒ {S}_{2} = \mathop{∑}_{r = 1}^{7}\left(2r\right)\left(16{r}^{2} + 1 - 8r\right)$

$⇒ {S}_{2} = 32\mathop{∑}_{r = 1}^{7}{r}^{3} + 2\mathop{∑}_{r = 1}^{7}r - 16\mathop{∑}_{r = 1}^{7}{r}^{2}$

$⇒ {S}_{2} = 32{\left(\dfrac{7 \times 8}{2}\right)}^{2} + 2\left(\dfrac{7 \times 8}{2}\right) - 16\left(\dfrac{7 \times 8 \times 15}{6}\right)$

$⇒ {S}_{2} = 32 \times 784 + 56 - 16 \times 140 = 22904$

Hence, $S = {S}_{1} - 2{S}_{2} = 52760 - 2 \times 22904 = 6952$